How to Evaluate Trigonometric Function? (+FREE Worksheet!)

Related Topics

• How to Solve Angles and Angle Measure
• How to Solve Coterminal Angles and Reference Angles
• How to Find Missing Sides and Angles of a Right Triangle
• How to Solve Trig Ratios of General Angles

Step by step guide to Evaluating Trigonometric Function

• Find the reference angle. (It is the smallest angle that you can make from the terminal side of an angle with the $x$-axis.)
• Find the trigonometric function of the reference angle.

Evaluating Trigonometric Function – Example 1:

Find the exact value of the trigonometric function. $tan$$\frac{4π}{3}$

Solution:

Rewrite the angles for an $\frac{4π}{3}$:
$tan$ $\frac{4π}{3}=tan \frac{3π+π}{3}=tan⁡(π+\frac{1}{3} π)$
Use the periodicity of $tan$: $tan$$(x+π .k)= tan (x)$
$tan$$⁡(π+\frac{1}{3} π)=$ $tan$$⁡(\frac{1}{3} π)=\sqrt{3}$

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Evaluating Trigonometric Function – Example 2:

Find the exact value of the trigonometric function. $cos$ $270^\circ$

Solution:

Write $cos$ $(270^\circ)$ as $cos$ $(180^\circ+90^\circ)$. Recall that $cos⁡180^\circ=-1,cos ⁡90^\circ =0$
The reference angle of $270^\circ$ is $90^\circ$. Therefore, $cos$ $270^\circ=0$

Evaluating Trigonometric Function – Example 3:

Find the exact value of the trigonometric function. $cos$ $225^\circ$

Solution:

Write $cos$ $(225^\circ)$ as $cos$ $(180^\circ+45^\circ)$. Recall that $cos$$⁡180^\circ=-1$, $cos$$⁡45^\circ =\frac{\sqrt{2}}{2}$
$225^\circ$ is in the third quadrant and $cos$ is negative in the quadrant $3$. The reference angle of $225^\circ$ is $45^\circ$. Therefore, $cos$ $225^\circ=-\frac{\sqrt{2}}{2}$

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Evaluating Trigonometric Function – Example 4:

Find the exact value of the trigonometric function. $tan$ $\frac{7π}{6}$

Solution:

Rewrite the angles for $tan$ $\frac{7π}{6}$:
$tan$ $\frac{7π}{6}=$ $tan$ $(\frac{6π+π}{6})=tan⁡(π+\frac{1}{6} π)$
Use the periodicity of $tan$: $tan$$(x+π .k)=$ $tan$$(x)$
$tan⁡(π+\frac{1}{6} π)=$ $tan$$⁡(\frac{1}{6} π)=\frac{\sqrt{3}}{3}$

Exercises for Evaluating Trigonometric Function

Find the exact value of each trigonometric function.

• $\color{blue}{cot \ -495^\circ=}$
• $\color{blue}{tan \ 405^\circ=}$
• $\color{blue}{cot \ 390^\circ=}$
• $\color{blue}{cos \ -300^\circ=}$
• $\color{blue}{cot \ -210^\circ=}$
• $\color{blue}{tan \ \frac{7π}{6}=}$

Download Evaluating Trigonometric Function Worksheet

• $\color{blue}{1}$
• $\color{blue}{1}$
• $\color{blue}{\sqrt{3}}$
• $\color{blue}{\frac{1}{2}}$
• $\color{blue}{- \sqrt{3} }$
• $\color{blue}{\frac{ \sqrt{3} }{3}}$

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