Trigonometric Functions in the Cartesian Plane

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

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˚

Videos

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.

Evaluating Trigonometric Functions Using the Reference Angle, Example 1
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).

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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