Wize High School Grade 11 Math Textbook > Trigonometry

Trig Ratios for Angles Between 0° and 360°

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Trig Ratios for Angles Between 0° and 360°
Standard Position
An angle in standard position starts at the initial arm (flat along the positive x-axis) and rotates counter-clockwise around the origin to a terminal arm.

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Angles Greater Than 90°\colorOne{90\degree}90°
The angle in standard position is called the principal angle (between 000 and 360°360\degree360°).
The principle angle can be negative if the angle is measured clockwise.

When the angle is greater than 90°90\degree90°, the angle between the terminal arm and the x-axis is called the related acute angle (or reference angle).

Wize Tip
Trig ratios of a principal angle are the same as the trig ratios of the reference angle, except they may be negative.

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CAST Rule
Trig ratios of a principal angle are the same as the trig ratios of the reference angle, except they may be negative.
Ex. Given a principal angle of 135°135\degree135°, the reference angle is 45°45\degree45°. To calculate sin⁡(135°)\sin(135\degree)sin(135°), we can instead calculate sin⁡(45°)\sin(45\degree)sin(45°)!

We can decide whether we should negate the result of using the reference angle by applying the CAST rule.
Looking at the quadrant where the terminal arm lies:

C - Cosine Positive (Quadrant IV)

A - All Positive (Quadrant I)

S - Sine Positive (Quadrant II)

T - Tangent Positive (Quadrant III)

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Example: Trig Ratios for Angles Between 0° and 360°
Evaluate sin⁡135°\sin135\degreesin135° as an exact value.

To evaluate sin⁡135°\sin135\degreesin135°, we find the reference angle. 
Together, these angles must make 180°180\degree180°, so the reference angle is 180°−135°=45°180\degree -135\degree = \colorbox{yellow}{45\degree}180°−135°=45°​.

Now, we know that either sin⁡135°=sin⁡45°\sin135\degree = \sin45\degreesin135°=sin45° or sin⁡135°=−sin⁡45°\sin135\degree = -\sin45\degreesin135°=−sin45°.
By the CAST rule, the terminal arm is in quadrant 2 (S), so the Sine of this angle is positive.

Therefore, sin⁡135°=sin⁡45°\sin135\degree = \sin45\degreesin135°=sin45°.
Filling in the special 45°- 45°- 90°45\degree \text{- }45\degree\text{- }90\degree45°- 45°- 90° triangle, we see that:
sin⁡135°=sin⁡45°=12=22\sin135\degree = \sin45\degree=\dfrac{1}{\sqrt2}= \boxed{\dfrac{\sqrt2}{2}}sin135°=sin45°=2​1​=22​​​

Practice: Trig Ratios for Angles Between 0° and 360°
Match each reference angle with the corresponding principal angle.

A.

β=15°\beta=15\degreeβ=15°

B.

β=60°\beta=60\degreeβ=60°

C.

β=45°\beta=45\degreeβ=45°

D.

β=30°\beta=30\degreeβ=30°

θ=150°\theta=150\degreeθ=150°

θ=225°\theta=225\degreeθ=225°

θ=120°\theta=120\degreeθ=120°

θ=345°\theta=345\degreeθ=345°

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Practice: Trig Ratios for Angles Between 0° and 360°
Complete the following table, given the reference angle (related acute angle) and the quadrant of the terminal arm.

You'll need to:
find the principal angle, and
determine whether or not the Cosine of the principal angle is positive

Reference Angle	Quadrant	Principal Angle	Cosine Positive?
12°	IV
81°	I
29°	II
7°	III

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Practice: Trig Ratios for Angles Between 0o and 360o
Evaluate the exact values of the following trig ratios without using a calculator.

a) sin⁡30°\sin 30\degreesin30°

b) cos⁡120°\cos 120\degreecos120°

c) tan⁡150°\tan150\degreetan150°

d) sin⁡225°\sin225\degreesin225°

e) tan⁡225°\tan225\degreetan225°

f) cos⁡(−30°)\cos(-30\degree)cos(−30°)

I don't know