Symmetry and Periodicity of Trigonometric Functions: CCSS.Math.Content.HSF.TF.B.4 - Common Core: High School - Functions

Card 0 of 12

Question

Find the following trigonometric exact value.

$\cos\left(\frac{\pi}{6} \right )$

Answer

This question tests one's ability to recognize and use the unit circle to calculate the exact trigonometric value.

For the purpose of Common Core Standards, "Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions." falls within the Cluster A of "Extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF-TF.A.4).

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.

Step 1: Recall the unit circle.

Step 2: Identify the coordinate pair that represents the extension of $\theta = \frac{\pi}{6}$ .

$\theta = \frac{\pi}{6}\Rightarrow \left(\frac{\sqrt{3}}{2}, \frac{1}{2} \right )$

Step 3: Calculate the exact value of $\cos\left(\frac{\pi}{6} \right )$ .

Recall that,

$(x,y)=(\cos(\theta),\sin(\theta))$

therefore,

$\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)=\left(\cos\left(\frac{\pi}{6}\right),\sin\left(\frac{\pi}{6}\right)\right)$

$\cos\left(\frac{\pi}{6} \right )=\frac{\sqrt{3}}{2}$ .

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Question

Find the following trigonometric exact value.

$\sin\left(\frac{\pi}{6} \right )$

Answer

This question tests one's ability to recognize and use the unit circle to calculate the exact trigonometric value.

For the purpose of Common Core Standards, "Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions." falls within the Cluster A of "Extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF-TF.A.4).

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.

Step 1: Recall the unit circle.

Step 2: Identify the coordinate pair that represents the extension of $\theta = \frac{\pi}{6}$ .

$\theta = \frac{\pi}{6}\Rightarrow \left(\frac{\sqrt{3}}{2}, \frac{1}{2} \right )$

Step 3: Calculate the exact value of $\sin\left(\frac{\pi}{6} \right )$ .

Recall that,

$(x,y)=(\cos(\theta),\sin(\theta))$

therefore,

$\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)=\left(\cos\left(\frac{\pi}{6}\right),\sin\left(\frac{\pi}{6}\right)\right)$

$\sin\left(\frac{\pi}{6} \right )=\frac{1}{2}$ .

Compare your answer with the correct one above

Question

Find the following trigonometric exact value.

$\cos\left(\frac{5\pi}{6} \right )$

Answer

This question tests one's ability to recognize and use the unit circle to calculate the exact trigonometric value.

For the purpose of Common Core Standards, "Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions." falls within the Cluster A of "Extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF-TF.A.4).

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.

Step 1: Recall the unit circle.

Step 2: Identify the coordinate pair that represents the extension of $\theta = \frac{5\pi}{6}$ .

$\theta = \frac{5\pi}{6}\Rightarrow \left(\frac{-\sqrt{3}}{2}, \frac{1}{2} \right )$

Step 3: Calculate the exact value of $\cos\left(\frac{5\pi}{6} \right )$ .

Recall that,

$(x,y)=(\cos(\theta),\sin(\theta))$

therefore,

$\left(-\frac{\sqrt{3}}{2},\frac{1}{2}\right)=\left(\cos\left(\frac{5\pi}{6}\right),\sin\left(\frac{5\pi}{6}\right)\right)$

$\cos\left(\frac{5\pi}{6} \right )=-\frac{\sqrt{3}}{2}$ .

Compare your answer with the correct one above

Question

Find the following trigonometric exact value.

$\sin\left(\frac{5\pi}{6} \right )$

Answer

This question tests one's ability to recognize and use the unit circle to calculate the exact trigonometric value.

For the purpose of Common Core Standards, "Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions." falls within the Cluster A of "Extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF-TF.A.4).

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.

Step 1: Recall the unit circle.

Step 2: Identify the coordinate pair that represents the extension of $\theta = \frac{5\pi}{6}$ .

$\theta = \frac{5\pi}{6}\Rightarrow \left(\frac{-\sqrt{3}}{2}, \frac{1}{2} \right )$

Step 3: Calculate the exact value of $\sin\left(\frac{5\pi}{6} \right )$ .

Recall that,

$(x,y)=(\cos(\theta),\sin(\theta))$

therefore,

$\left(-\frac{\sqrt{3}}{2},\frac{1}{2}\right)=\left(\cos\left(\frac{5\pi}{6}\right),\sin\left(\frac{5\pi}{6}\right)\right)$

$\sin\left(\frac{5\pi}{6} \right )=\frac{1}{2}$ .

Compare your answer with the correct one above

Question

Find the following trigonometric exact value.

$\cos\left(\frac{7\pi}{6} \right )$

Answer

This question tests one's ability to recognize and use the unit circle to calculate the exact trigonometric value.

For the purpose of Common Core Standards, "Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions." falls within the Cluster A of "Extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF-TF.A.4).

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.

Step 1: Recall the unit circle.

Step 2: Identify the coordinate pair that represents the extension of $\theta = \frac{7\pi}{6}$ .

$\theta = \frac{7\pi}{6}\Rightarrow \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2} \right )$

Step 3: Calculate the exact value of $\cos\left(\frac{7\pi}{6} \right )$ .

Recall that,

$(x,y)=(\cos(\theta),\sin(\theta))$

therefore,

$\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right)=\left(\cos\left(\frac{7\pi}{6}\right),\sin\left(\frac{7\pi}{6}\right)\right)$

$\cos\left(\frac{7\pi}{6} \right )=-\frac{\sqrt{3}}{2}$ .

Compare your answer with the correct one above

Question

Find the following trigonometric exact value.

$\sin\left(\frac{7\pi}{6} \right )$

Answer

This question tests one's ability to recognize and use the unit circle to calculate the exact trigonometric value.

For the purpose of Common Core Standards, "Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions." falls within the Cluster A of "Extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF-TF.A.4).

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.

Step 1: Recall the unit circle.

Step 2: Identify the coordinate pair that represents the extension of $\theta = \frac{7\pi}{6}$ .

$\theta = \frac{7\pi}{6}\Rightarrow \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2} \right )$

Step 3: Calculate the exact value of $\sin\left(\frac{7\pi}{6} \right )$ .

Recall that,

$(x,y)=(\cos(\theta),\sin(\theta))$

therefore,

$\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right)=\left(\cos\left(\frac{7\pi}{6}\right),\sin\left(\frac{7\pi}{6}\right)\right)$

$\sin\left(\frac{7\pi}{6} \right )=-\frac{1}{2}$ .

Compare your answer with the correct one above

Question

Find the following trigonometric exact value.

$\cos\left(\frac{11\pi}{6} \right )$

Answer

This question tests one's ability to recognize and use the unit circle to calculate the exact trigonometric value.

For the purpose of Common Core Standards, "Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions." falls within the Cluster A of "Extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF-TF.A.4).

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.

Step 1: Recall the unit circle.

Step 2: Identify the coordinate pair that represents the extension of $\theta = \frac{11\pi}{6}$ .

$\theta = \frac{11\pi}{6}\Rightarrow \left(\frac{\sqrt{3}}{2}, -\frac{1}{2} \right )$

Step 3: Calculate the exact value of $\cos\left(\frac{11\pi}{6} \right )$ .

Recall that,

$(x,y)=(\cos(\theta),\sin(\theta))$

therefore,

$\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)=\left(\cos\left(\frac{11\pi}{6}\right),\sin\left(\frac{11\pi}{6}\right)\right)$

$\cos\left(\frac{11\pi}{6} \right )=\frac{\sqrt{3}}{2}$ .

Compare your answer with the correct one above

Question

Find the following trigonometric exact value.

$\sin\left(\frac{11\pi}{6} \right )$

Answer

This question tests one's ability to recognize and use the unit circle to calculate the exact trigonometric value.

For the purpose of Common Core Standards, "Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions." falls within the Cluster A of "Extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF-TF.A.4).

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.

Step 1: Recall the unit circle.

Step 2: Identify the coordinate pair that represents the extension of $\theta = \frac{11\pi}{6}$ .

$\theta = \frac{11\pi}{6}\Rightarrow \left(\frac{\sqrt{3}}{2}, -\frac{1}{2} \right )$

Step 3: Calculate the exact value of $\sin\left(\frac{11\pi}{6} \right )$ .

Recall that,

$(x,y)=(\cos(\theta),\sin(\theta))$

therefore,

$\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)=\left(\cos\left(\frac{11\pi}{6}\right),\sin\left(\frac{11\pi}{6}\right)\right)$

$\sin\left(\frac{11\pi}{6} \right )=-\frac{1}{2}$ .

Compare your answer with the correct one above

Question

Find the following trigonometric exact value.

$\cos\left(\frac{\pi}{4} \right )$

Answer

This question tests one's ability to recognize and use the unit circle to calculate the exact trigonometric value.

For the purpose of Common Core Standards, "Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions." falls within the Cluster A of "Extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF-TF.A.4).

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.

Step 1: Recall the unit circle.

Step 2: Identify the coordinate pair that represents the extension of $\theta = \frac{\pi}{4}$ .

$\theta = \frac{\pi}{4}\Rightarrow \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right )$

Step 3: Calculate the exact value of $\cos\left(\frac{\pi}{4} \right )$ .

Recall that,

$(x,y)=(\cos(\theta),\sin(\theta))$

therefore,

$\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)=\left(\cos\left(\frac{\pi}{4}\right),\sin\left(\frac{\pi}{4}\right)\right)$

$\cos\left(\frac{\pi}{4} \right )=\frac{1}{\sqrt{2}}$ .

Compare your answer with the correct one above

Question

Find the following trigonometric exact value.

$\sin\left(\frac{\pi}{4} \right )$

Answer

This question tests one's ability to recognize and use the unit circle to calculate the exact trigonometric value.

For the purpose of Common Core Standards, "Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions." falls within the Cluster A of "Extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF-TF.A.4).

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.

Step 1: Recall the unit circle.

Step 2: Identify the coordinate pair that represents the extension of $\theta = \frac{\pi}{4}$ .

$\theta = \frac{\pi}{4}\Rightarrow \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right )$

Step 3: Calculate the exact value of $\sin\left(\frac{\pi}{4} \right )$ .

Recall that,

$(x,y)=(\cos(\theta),\sin(\theta))$

therefore,

$\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)=\left(\cos\left(\frac{\pi}{4}\right),\sin\left(\frac{\pi}{4}\right)\right)$

$\sin\left(\frac{\pi}{4} \right )=\frac{1}{\sqrt{2}}$ .

Compare your answer with the correct one above

Question

Find the following trigonometric exact value.

$\cos\left(\frac{3\pi}{4} \right )$

Answer

This question tests one's ability to recognize and use the unit circle to calculate the exact trigonometric value.

For the purpose of Common Core Standards, "Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions." falls within the Cluster A of "Extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF-TF.A.4).

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.

Step 1: Recall the unit circle.

Step 2: Identify the coordinate pair that represents the extension of $\theta = \frac{3\pi}{4}$ .

$\theta = \frac{3\pi}{4}\Rightarrow \left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right )$

Step 3: Calculate the exact value of $\cos\left(\frac{3\pi}{4} \right )$ .

Recall that,

$(x,y)=(\cos(\theta),\sin(\theta))$

therefore,

$\left(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)=\left(\cos\left(\frac{3\pi}{4}\right),\sin\left(\frac{3\pi}{4}\right)\right)$

$\cos\left(\frac{3\pi}{4} \right )=-\frac{1}{\sqrt{2}}$ .

Compare your answer with the correct one above

Question

Find the following trigonometric exact value.

$\sin\left(\frac{3\pi}{4} \right )$

Answer

This question tests one's ability to recognize and use the unit circle to calculate the exact trigonometric value.

For the purpose of Common Core Standards, "Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions." falls within the Cluster A of "Extend the domain of trigonometric functions using the unit circle" concept (CCSS.MATH.CONTENT.HSF-TF.A.4).

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.

Step 1: Recall the unit circle.

Step 2: Identify the coordinate pair that represents the extension of $\theta = \frac{3\pi}{4}$ .

$\theta = \frac{3\pi}{4}\Rightarrow \left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right )$

Step 3: Calculate the exact value of $\sin\left(\frac{3\pi}{4} \right )$ .

Recall that,

$(x,y)=(\cos(\theta),\sin(\theta))$

therefore,

$\left(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)=\left(\cos\left(\frac{3\pi}{4}\right),\sin\left(\frac{3\pi}{4}\right)\right)$

$\sin\left(\frac{3\pi}{4} \right )=\frac{1}{\sqrt{2}}$ .

Compare your answer with the correct one above

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