# Common Core: High School - Functions : Use Special Triangles to Determine Trigonometric Functions: CCSS.Math.Content.HSF-TF.A.3

## Example Questions

### Example Question #1 : Use Special Triangles To Determine Trigonometric Functions: Ccss.Math.Content.Hsf Tf.A.3

Calculate sine, cosine, and tangent of the following angle.

Explanation:

This question tests one's ability to understand the connection between special triangles (30-60-90 and 45-45-90), trigonometric functions related to them (sine, cosine, tangent), and the corresponding degree/radian on the unit circle. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each  pair that lies on the circle can be found by creating a right triangle that has a base on the -axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the -axis.

For the purpose of Common Core Standards, understanding "use special triangles to determine geometrically the values of sine, cosine, tangent for " , falls within the Cluster A of "extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Possible approaches to solving this problem:

I. Plot the angle on the unit circle to find the .

II. Identify the special triangle that corresponds to the angle.

III. Using memorization of the unit circle.

For this particular question let's us approach II.

Step 1: Simplify the angle if possible.

Step 2: Identify the special right triangle that corresponds to the angle.

Let's convert the angle to degrees to more easily see the special right triangle that it creates.

Therefore the special triangle that is created with  is 30-60-90 degree triangle. The angle which is made at the origin will be 60 degrees and the length of the segment from the origin to the edge of the unit circle is one by definition.

Step 3: Calculate the side lengths of the special triangle.

From here, recall that the special 30-60-90 degree triangle has the following side identities.

Substituting in one for the hypotenuse, the short and long leg of the triangle can be found.

It is important to remember that the short side of the triangle is on the -axis and adjacent to . The long side is the vertical line that is parallel to the -axis and opposite of .

Step 4: Calculate sine, cosine, and tangent.

Recall the following trigonometric identities.

From Step 2,

Substituting the values found in Step 2 into the above identities is as follows.

### Example Question #2 : Use Special Triangles To Determine Trigonometric Functions: Ccss.Math.Content.Hsf Tf.A.3

Calculate sine of the following angle.

Explanation:

This question tests one's ability to understand the connection between special triangles (30-60-90 and 45-45-90), trigonometric functions related to them (sine, cosine, tangent), and the corresponding degree/radian on the unit circle. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each  pair that lies on the circle can be found by creating a right triangle that has a base on the -axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the -axis.

For the purpose of Common Core Standards, understanding "use special triangles to determine geometrically the values of sine, cosine, tangent for " , falls within the Cluster A of "extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Possible approaches to solving this problem:

I. Plot the angle on the unit circle to find the .

II. Identify the special triangle that corresponds to the angle.

III. Using memorization of the unit circle.

For this particular question let's us approach II.

Step 1: Simplify the angle if possible.

Step 2: Identify the special right triangle that corresponds to the angle.

Let's convert the angle to degrees to more easily see the special right triangle that it creates.

Therefore the special triangle that is created with  is 45-45-90 degree triangle. The angle which is made at the origin will be 45 degrees and the length of the segment from the origin to the edge of the unit circle is one by definition.

Step 3: Calculate the side lengths of the special triangle.

From here, recall that the special 45-45-90 degree triangle has the following side identities.

Step 4: Calculate sine.

Recall the following trigonometric identities.

### Example Question #3 : Use Special Triangles To Determine Trigonometric Functions: Ccss.Math.Content.Hsf Tf.A.3

Calculate cosine of the following angle.

Explanation:

This question tests one's ability to understand the connection between special triangles (30-60-90 and 45-45-90), trigonometric functions related to them (sine, cosine, tangent), and the corresponding degree/radian on the unit circle. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each  pair that lies on the circle can be found by creating a right triangle that has a base on the -axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the -axis.

For the purpose of Common Core Standards, understanding "use special triangles to determine geometrically the values of sine, cosine, tangent for " , falls within the Cluster A of "extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Possible approaches to solving this problem:

I. Plot the angle on the unit circle to find the .

II. Identify the special triangle that corresponds to the angle.

III. Using memorization of the unit circle.

For this particular question let's us approach II.

Step 1: Simplify the angle if possible.

Step 2: Identify the special right triangle that corresponds to the angle.

Let's convert the angle to degrees to more easily see the special right triangle that it creates.

Therefore the special triangle that is created with  is 45-45-90 degree triangle. The angle which is made at the origin will be 45 degrees and the length of the segment from the origin to the edge of the unit circle is one by definition.

Step 3: Calculate the side lengths of the special triangle.

From here, recall that the special 45-45-90 degree triangle has the following side identities.

Step 4: Calculate cosine.

Recall the following trigonometric identities.

### Example Question #4 : Use Special Triangles To Determine Trigonometric Functions: Ccss.Math.Content.Hsf Tf.A.3

Calculate tangent of the following angle.

Explanation:

This question tests one's ability to understand the connection between special triangles (30-60-90 and 45-45-90), trigonometric functions related to them (sine, cosine, tangent), and the corresponding degree/radian on the unit circle. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each  pair that lies on the circle can be found by creating a right triangle that has a base on the -axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the -axis.

For the purpose of Common Core Standards, understanding "use special triangles to determine geometrically the values of sine, cosine, tangent for " , falls within the Cluster A of "extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Possible approaches to solving this problem:

I. Plot the angle on the unit circle to find the .

II. Identify the special triangle that corresponds to the angle.

III. Using memorization of the unit circle.

For this particular question let's us approach II.

Step 1: Simplify the angle if possible.

Step 2: Identify the special right triangle that corresponds to the angle.

Let's convert the angle to degrees to more easily see the special right triangle that it creates.

Therefore the special triangle that is created with  is 45-45-90 degree triangle. The angle which is made at the origin will be 45 degrees and the length of the segment from the origin to the edge of the unit circle is one by definition.

Step 3: Calculate the side lengths of the special triangle.

From here, recall that the special 45-45-90 degree triangle has the following side identities.

Step 4: Calculate tangent.

Recall the following trigonometric identities.

Recall that when dividing fractions, one can multiply the numerator by the reciprocal of the denominator.

### Example Question #5 : Use Special Triangles To Determine Trigonometric Functions: Ccss.Math.Content.Hsf Tf.A.3

Calculate tangent of the following angle.

Explanation:

This question tests one's ability to understand the connection between special triangles (30-60-90 and 45-45-90), trigonometric functions related to them (sine, cosine, tangent), and the corresponding degree/radian on the unit circle. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each  pair that lies on the circle can be found by creating a right triangle that has a base on the -axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the -axis.

For the purpose of Common Core Standards, understanding "use special triangles to determine geometrically the values of sine, cosine, tangent for " , falls within the Cluster A of "extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Possible approaches to solving this problem:

I. Plot the angle on the unit circle to find the .

II. Identify the special triangle that corresponds to the angle.

III. Using memorization of the unit circle.

For this particular question let's us approach II.

Step 1: Simplify the angle if possible.

Step 2: Identify the special right triangle that corresponds to the angle.

Let's convert the angle to degrees to more easily see the special right triangle that it creates.

Find the reference angle by subtracting the angle from from 180 degrees.

This means that 135 degrees is a 45 degree angle that lies in the second quadrant. Recall that the second quadrant contains positive  values and negative  values.

Therefore the special triangle that is created with  is 45-45-90 degree triangle. The angle which is made at the origin will be 45 degrees and the length of the segment from the origin to the edge of the unit circle is one by definition.

Step 3: Calculate the side lengths of the special triangle.

From here, recall that the special 45-45-90 degree triangle has the following side identities.

Step 4: Calculate tangent.

Recall the following trigonometric identities.

Since tangent represents the  values over the  values on the coordinate grid, this means that tangent is negative for angles in quadrant two.

### Example Question #6 : Use Special Triangles To Determine Trigonometric Functions: Ccss.Math.Content.Hsf Tf.A.3

Calculate sine of the following angle.

Explanation:

This question tests one's ability to understand the connection between special triangles (30-60-90 and 45-45-90), trigonometric functions related to them (sine, cosine, tangent), and the corresponding degree/radian on the unit circle. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each  pair that lies on the circle can be found by creating a right triangle that has a base on the -axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the -axis.

For the purpose of Common Core Standards, understanding "use special triangles to determine geometrically the values of sine, cosine, tangent for " , falls within the Cluster A of "extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Possible approaches to solving this problem:

I. Plot the angle on the unit circle to find the .

II. Identify the special triangle that corresponds to the angle.

III. Using memorization of the unit circle.

For this particular question let's us approach II.

Step 1: Simplify the angle if possible.

Step 2: Identify the special right triangle that corresponds to the angle.

Let's convert the angle to degrees to more easily see the special right triangle that it creates.

Find the reference angle by subtracting the angle from from 180 degrees.

This means that 135 degrees is a 45 degree angle that lies in the second quadrant. Recall that the second quadrant contains positive  values and negative  values.

Therefore the special triangle that is created with  is 45-45-90 degree triangle. The angle which is made at the origin will be 45 degrees and the length of the segment from the origin to the edge of the unit circle is one by definition.

Step 3: Calculate the side lengths of the special triangle.

From here, recall that the special 45-45-90 degree triangle has the following side identities.

Step 4: Calculate sine.

Recall the following trigonometric identities.

Since sine represents the  values on the coordinate grid, this means that sine remains positive for angles in quadrant two.

### Example Question #7 : Use Special Triangles To Determine Trigonometric Functions: Ccss.Math.Content.Hsf Tf.A.3

Calculate cosine of the following angle.

Explanation:

This question tests one's ability to understand the connection between special triangles (30-60-90 and 45-45-90), trigonometric functions related to them (sine, cosine, tangent), and the corresponding degree/radian on the unit circle. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each  pair that lies on the circle can be found by creating a right triangle that has a base on the -axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the -axis.

For the purpose of Common Core Standards, understanding "use special triangles to determine geometrically the values of sine, cosine, tangent for " , falls within the Cluster A of "extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Possible approaches to solving this problem:

I. Plot the angle on the unit circle to find the .

II. Identify the special triangle that corresponds to the angle.

III. Using memorization of the unit circle.

For this particular question let's us approach II.

Step 1: Simplify the angle if possible.

Step 2: Identify the special right triangle that corresponds to the angle.

Let's convert the angle to degrees to more easily see the special right triangle that it creates.

Find the reference angle by subtracting the angle from from 180 degrees.

This means that 135 degrees is a 45 degree angle that lies in the second quadrant. Recall that the second quadrant contains positive  values and negative  values.

Therefore the special triangle that is created with  is 45-45-90 degree triangle. The angle which is made at the origin will be 45 degrees and the length of the segment from the origin to the edge of the unit circle is one by definition.

Step 3: Calculate the side lengths of the special triangle.

From here, recall that the special 45-45-90 degree triangle has the following side identities.

Step 4: Calculate cosine.

Recall the following trigonometric identities.

Since cosine represents the  values on the coordinate grid, this means that cosine is negative for angles in quadrant two.

### Example Question #8 : Use Special Triangles To Determine Trigonometric Functions: Ccss.Math.Content.Hsf Tf.A.3

Calculate tangent of the following angle.

Explanation:

This question tests one's ability to understand the connection between special triangles (30-60-90 and 45-45-90), trigonometric functions related to them (sine, cosine, tangent), and the corresponding degree/radian on the unit circle. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each  pair that lies on the circle can be found by creating a right triangle that has a base on the -axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the -axis.

For the purpose of Common Core Standards, understanding "use special triangles to determine geometrically the values of sine, cosine, tangent for " , falls within the Cluster A of "extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Possible approaches to solving this problem:

I. Plot the angle on the unit circle to find the .

II. Identify the special triangle that corresponds to the angle.

III. Using memorization of the unit circle.

For this particular question let's us approach II.

Step 1: Simplify the angle if possible.

Step 2: Identify the special right triangle that corresponds to the angle.

Let's convert the angle to degrees to more easily see the special right triangle that it creates.

Find the reference angle by subtracting 180 degrees from the angle.

This means that 225 degrees is a 45 degree angle that lies in the third quadrant. Recall that the third quadrant contains negative  values and negative  values.

Therefore the special triangle that is created with  is 45-45-90 degree triangle. The angle which is made at the origin will be 45 degrees and the length of the segment from the origin to the edge of the unit circle is one by definition.

Step 3: Calculate the side lengths of the special triangle.

From here, recall that the special 45-45-90 degree triangle has the following side identities.

Step 4: Calculate tangent.

Recall the following trigonometric identities.

Since tangent represents the  values over the  values on the coordinate grid, this means that tangent is positive for angles in quadrant three.

### Example Question #9 : Use Special Triangles To Determine Trigonometric Functions: Ccss.Math.Content.Hsf Tf.A.3

Calculate sine of the following angle.

Explanation:

This question tests one's ability to understand the connection between special triangles (30-60-90 and 45-45-90), trigonometric functions related to them (sine, cosine, tangent), and the corresponding degree/radian on the unit circle. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each  pair that lies on the circle can be found by creating a right triangle that has a base on the -axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the -axis.

For the purpose of Common Core Standards, understanding "use special triangles to determine geometrically the values of sine, cosine, tangent for " , falls within the Cluster A of "extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Possible approaches to solving this problem:

I. Plot the angle on the unit circle to find the .

II. Identify the special triangle that corresponds to the angle.

III. Using memorization of the unit circle.

For this particular question let's us approach II.

Step 1: Simplify the angle if possible.

Step 2: Identify the special right triangle that corresponds to the angle.

Let's convert the angle to degrees to more easily see the special right triangle that it creates.

Find the reference angle by subtracting 180 degrees from the angle.

This means that 225 degrees is a 45 degree angle that lies in the third quadrant. Recall that the third quadrant contains negative  values and negative  values.

Therefore the special triangle that is created with  is 45-45-90 degree triangle. The angle which is made at the origin will be 45 degrees and the length of the segment from the origin to the edge of the unit circle is one by definition.

Step 3: Calculate the side lengths of the special triangle.

From here, recall that the special 45-45-90 degree triangle has the following side identities.

Step 4: Calculate tangent.

Recall the following trigonometric identities.

### Example Question #10 : Use Special Triangles To Determine Trigonometric Functions: Ccss.Math.Content.Hsf Tf.A.3

Calculate cosine of the following angle.

Explanation:

This question tests one's ability to understand the connection between special triangles (30-60-90 and 45-45-90), trigonometric functions related to them (sine, cosine, tangent), and the corresponding degree/radian on the unit circle. It relies on the understanding that the measure of an angle can be described using the properties of the unit circle and arc length. Furthermore, questions involving concepts dealing with trigonometry use the foundation of corresponding identities with angles that come from the unit circle. Recall that the unit circle is a circle that has a radius of one and is centered at the origin. Each  pair that lies on the circle can be found by creating a right triangle that has a base on the -axis and a hypotenuse that is from the center of the circle to the desired point on the edge of the circle. The height of the triangle is the vertical line that connects the point on the circle's edge to the -axis.

For the purpose of Common Core Standards, understanding "use special triangles to determine geometrically the values of sine, cosine, tangent for " , falls within the Cluster A of "extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF.TF.A).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Possible approaches to solving this problem:

I. Plot the angle on the unit circle to find the .

II. Identify the special triangle that corresponds to the angle.

III. Using memorization of the unit circle.

For this particular question let's us approach II.

Step 1: Simplify the angle if possible.

Step 2: Identify the special right triangle that corresponds to the angle.

Let's convert the angle to degrees to more easily see the special right triangle that it creates.

Find the reference angle by subtracting 180 degrees from the angle.

This means that 225 degrees is a 45 degree angle that lies in the third quadrant. Recall that the third quadrant contains negative  values and negative  values.

Therefore the special triangle that is created with  is 45-45-90 degree triangle. The angle which is made at the origin will be 45 degrees and the length of the segment from the origin to the edge of the unit circle is one by definition.

Step 3: Calculate the side lengths of the special triangle.

From here, recall that the special 45-45-90 degree triangle has the following side identities.

Step 4: Calculate tangent.

Recall the following trigonometric identities.
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