In these lessons, we will look at Trigonometric Functions for any angle in the Cartesian Plane by using the reference angle.

Related Topics:

More Lessons on Trigonometry

Steps to solving trigonometric functions for any angle

**Step 1:** Find the Reference Angle, which is always acute

**Step 2: **Find Trig Function Value for the reference angle

**Step 3: **Determine the Sign (positive or negative) of the trig function based on the quadrant

* Example:*

Find

a) sin 120°

b) cos 150°

c) tan 210°

d) csc 300°

* Solution: *

a) sin 120°

**Step 1: **Find the reference angle

180° – 120° = 60°

**Step 2: **Find Trig Function Value for the reference angle

sin 60° = 0.866

**Step 3:** Determine the Sign (positive or negative) of the trig function based on the quadrant

120° is in the second quadrant, where sin is positive.

So, sin 120° = sin 60° = 0.866

b) cos 150°

**Step 1:** Find the reference angle

180° – 150° = 30°

**Step 2:** Find Trig Function Value for the reference angle

cos 30° = 0.866

**Step 3: **Determine the Sign (positive or negative) of the trig function based on the quadrant

150° is in the second quadrant, where cos is negative

So, cos 150° = –cos 30° = –0.866

c) tan 210°

**Step 1: **Find the reference angle

210° – 180° = 30°

**Step 2: **Find Trig Function Value for the reference angle

tan 30° = 0.5774

**Step 3:** Determine the Sign (positive or negative) of the trig function based on the quadrant

210° is in the third quadrant, where tan is positive

So, tan 210° = tan 30° = 0.5774

d) csc 300°

**Step 1: **Find the reference angle

360° – 300° = 60°

**Step 2:** Find Trig Function Value for the reference angle

csc 60° = 1.155

**Step 3: **Determine the Sign (positive or negative) of the trig function based on the quadrant

300° is in the fourth quadrant, where csc is negative

So, csc 300° = –csc 60° = –1.155

* Example: *

Given that sin 56˚ = 0.83 and cos 56˚ = 0.56, find the value of

2 sin 304˚ + cos 124˚

* Solution*

Reference angle for 304˚ = (360˚ – 304˚) = 56˚

sin 304˚ = – (sin 56˚) = –0.83

Reference angle for 124˚ = (180˚ – 124˚) = 56˚

cos 124˚ = – (cos 56˚) = –0.56

2 sin 304˚ + cos 124˚ = 2 (–0.83) + (–0.56) = **–2.22**

* Example: *

Given that 0˚ ≤ * x* ≤ 360 ˚, find the angle *x* for each of the following:

a) sin *x* = –0.6691

b) cos *x* = 0.2079

c) tan *x* = –1.4281

* Solution: *

a) sin *x* = –0.6691

reference angle = sin -1 (0.6691)

reference angle = 42˚ (round to the nearest degree)

sin is negative in the quadrant III and IV

So, *x* = 180 + 42 = 222˚ or

*x* = 360 – 42 = 318˚

b) cos *x* = 0.2079

reference angle = cos -1 (0.2079)

reference angle = 78˚ (round to the nearest degree)

cos is positive in quadrant I and IV

So, *x* = 78˚ or

*x* = 360 – 78 = 282˚

c) tan *x* = –1.4281

reference angle = tan -1 (1.4281)

reference angle = 55˚ (round to the nearest degree)

tan is negative in quadrant II and IV

So, *x* = 180 – 55 = 125˚ or

*x* = 360 – 55 = 305˚

Trig Functions for any Angle: Positive or Negative, Greater than 360

Steps to solving trigonometric angles for any angle.

1) Find reference angle (always acute)

2) Find the Trig Function Value for the Angle (remember special angles)

3) Determine the sign (positive or negative) of Trig Function based on Quadrant.

This video gives a quick review of the unit circle in quadrant 1 and discusses how to use the reference angle to evaluate some trig functions.

Trig Functions for any Angle given in Radians

Solving Trig Functions exactly for any angle in radians.

Approximating Trig Functions in Radians using a calculator.

Evaluating Trigonometric Functions Using the Reference Angle, Example 2.

This video discusses how to use the reference angle to evaluate some more trig functions (in radians).

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