## How to Evaluate Each Trigonometric Function

### Trigonometric Functions

Trigonometric functions -also referred to as circular functions- are the functions of a triangle's angle. This means that these trig functions provide the relationship between the angles and sides of a triangle. Sine, cosine, tangent, cotangent, secant, and cosecant are the basic trigonometric functions.

### How to Evaluate Trigonometric Functions Step by Step

• Determine 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 reference angle's trigonometric function.

### Example

Evaluate $$tan(\frac{5π}{4})$$

Solution

Rewrite the angles for $$\frac{5π}{4}$$

$$tan(\frac{5π}{4}) \ = \ tan(π \ + \ \frac{π}{4})$$

Utilize the $$tan$$ periodicity: $$tan(θ \ + \ nπ) \ = \ tan(θ)$$

$$tan(π \ + \ \frac{π}{4}) \ = \ tan(\frac{π}{4}) \ = \ 1$$

• $$Sin(π \ - \ θ) \ = \ sin(θ)$$
• $$Cos(π \ - \ θ) \ = \ –cos(θ)$$
• $$tan(π \ - \ θ) \ = \ – tan(θ)$$
• $$Cosec(π \ - \ θ) \ = \ cosec(θ)$$
• $$Sec(π \ - \ θ) \ = \ -sec(θ)$$
• $$Cot(π \ - \ θ) \ = \ -cot(θ)$$

### Exercises for Evaluating Each Trigonometric Function

1) $$sin \ 120° \ =$$

2) $$tan \ 150° \ =$$

3) $$cot \ 330° \ =$$

4) $$tan \ 300° \ =$$

5) $$cos \ 225° \ =$$

6) $$sin \ \frac{3π}{2} \ =$$

7) $$cot \ \frac{11π}{6} \ =$$

8) $$sin \ \frac{2π}{3} \ =$$

9) $$cos \ \frac{7π}{6} \ =$$

10) $$tan \ \frac{9π}{4} \ =$$

1) $$sin \ 120° \ =$$

$$\color{red}{sin \ 120° \ = \ sin \ (180° \ - \ 60°) \ = \ sin \ 60° \ = \ \frac{\sqrt{3}}{2}}$$

2) $$tan \ 150° \ =$$

$$\color{red}{tan \ 150° \ = \ tan \ (180° \ - \ 30°) \ = \ -tan \ 30° \ = \ -\frac{\sqrt{3}}{3}}$$

3) $$cot \ 330° \ =$$

$$\color{red}{cot \ 330° \ = \ cot \ (360° \ - \ 30°) \ = \ -cot \ 30° \ = \ -\sqrt{3}}$$

4) $$tan \ 300° \ =$$

$$\color{red}{tan \ 300° \ = \ tan \ (300° \ - \ 60°) \ = \ -tan \ 60° \ = \ -\sqrt{3}}$$

5) $$cos \ 225° \ =$$

$$\color{red}{cos \ 225° \ = \ cos \ (180° \ + \ 45°) \ = \ -cos \ 45° \ = \ -\frac{\sqrt{2}}{2}}$$

6) $$sin \ \frac{3π}{2} \ =$$

$$\color{red}{sin \ \frac{3π}{2} \ = \ sin \ (π \ + \ \frac{π}{2}) \ = \ -sin \ \frac{π}{2} \ = -1}$$

7) $$cot \ \frac{11π}{6} \ =$$

$$\color{red}{cot \ \frac{11π}{6} \ = \ cot \ (2π \ - \ \frac{π}{6}) \ = \ -cot \ \frac{π}{6} \ = -\sqrt{3}}$$

8) $$sin \ \frac{2π}{3} \ =$$

$$\color{red}{sin \ \frac{2π}{3} \ = \ sin \ (π \ - \ \frac{π}{3}) \ = \ sin \ \frac{π}{3} \ = \frac{\sqrt{3}}{2}}$$

9) $$cos \ \frac{7π}{6} \ =$$

$$\color{red}{cos \ \frac{7π}{6} \ = \ cos \ (π \ + \ \frac{π}{6}) \ = \ -cos \ \frac{π}{6} \ = -\frac{\sqrt{3}}{2}}$$

10) $$tan \ \frac{9π}{4} \ =$$

$$\color{red}{tan \ \frac{9π}{4} \ = \ tan \ (2π \ + \ \frac{π}{4}) \ = \ tan \ \frac{π}{4} \ = \frac{\sqrt{2}}{2}}$$

## Evaluating Each Trigonometric Function Practice Quiz

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