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Wize High School Grade 12 Pre-Calculus Textbook > Trigonometric Ratios (Radians)

Trigonometric Ratios in Radians

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Trigonometric Ratios in Radians


Let P(x,y)P(x,y) be a point that moves around a circle with radius rr and center (0,0).(0,0).

For any position of P(x,y)P(x,y), the primary trigonometric ratios are:

cos⁡θ=xrsin⁡θ=yrtan⁡θ=yxr=x2+y2\begin{array}{rcl} \cos{\theta}&=&\displaystyle\frac{x}{r}\\\\ \sin{\theta}&=&\displaystyle\frac{y}{r}\\\\ \tan{\theta}&=&\displaystyle\frac{y}{x}\\\\ r&=&\sqrt{x^2+y^2} \end{array}
  • OP OP~ is the terminal arm.
  • The angle θ \theta~ is in standard position, which means that one side of the angle is fixed along the positive x-axis and the vertex is located at the origin.
  • The reference angle is the smallest angle that the terminal arm can make with the x-axis and can be determined in each quadrant.
  • Quadrant II: π−θ\pi-\theta
  • Quadrant III: π+θ\pi+\theta
  • Quadrant IV: 2π−θ2\pi-\theta
  • All coterminal angles to θ \theta~ can be found by θ±2πn  or  θ±360∘n,   n∈Z\theta\pm2\pi n~~\text{or}~~\theta\pm360^\circ{}n,~~~n\in\mathbb{Z}. Coterminal angles are angles that share the terminal side of an angle occupying the standard position.
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Example 1

Let θ=3π5\theta=\displaystyle\frac{3\pi}{5} . Determine:
  1. The reference angle
  2. 4 coterminal angles: 2 positive, 2 negative
a. Let's sketch where θ=3π5\theta=\displaystyle\frac{3\pi}{5} is.


Therefore, the reference angle is:

π−3π5=2π5\pi-\displaystyle\frac{3\pi}{5}=\boxed{\displaystyle\frac{2\pi}{5}}

b. 2 Positive Coterminal Angles:

3π5+2π=13π53π5+4π=23π5\begin{array}{rcl} \displaystyle\frac{3\pi}{5}+2\pi&=&\boxed{\displaystyle\frac{13\pi}{5}}\\\\ \displaystyle\frac{3\pi}{5}+4\pi&=&\boxed{\displaystyle\frac{23\pi}{5}} \end{array}

2 Negative Coterminal Angles:

3π5−2π=−7π53π5−4π=−17π5\begin{array}{rcl} \displaystyle\frac{3\pi}{5}-2\pi&=&\boxed{\displaystyle-\frac{7\pi}{5}}\\\\ \displaystyle\frac{3\pi}{5}-4\pi&=&\boxed{\displaystyle-\frac{17\pi}{5}} \end{array}

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Example 2

The point P(−2,3) P(-2, 3)~ is on the terminal arm of an angle θ.\theta.

Calculate the values of sin⁡θ, cos⁡θ, tan⁡θ\sin{\theta},~\cos{\theta},~\tan{\theta} and leave in exact form.

Find r‾:r=32+(−2)2r=13Find ratios‾:sin⁡θ=313cos⁡θ=−213tan⁡θ=−32\begin{array}{lrcl} \underline{\text{Find r}}:\\\\ &r&=&\sqrt{3^2+(-2)^2}\\\\ &r&=&\sqrt{13}\\\\\\ \underline{\text{Find ratios}}:\\\\ &\sin{\theta}&=&\displaystyle\frac{3}{\sqrt{13}}\\\\ &\cos{\theta}&=&\displaystyle\frac{-2}{\sqrt{13}}\\\\ &\tan{\theta}&=&-\displaystyle\frac{3}{2} \end{array}

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Example: Trigonometric Ratios in Radians


Each point P is on the terminal arm of an angle θ.\theta. Find cos⁡θ, sin⁡θ, tan⁡θ\cos{\theta},~\sin{\theta},~\tan{\theta}. Leave in exact form.
  1. P(−3,4)P(-3, 4)
  2. P(12,−32)P\Bigg(\displaystyle\frac{1}{2},\displaystyle-\frac{\sqrt{3}}{2}\Bigg)
1. P(−3,4)P(-3,4)

Find radius, r:

r=(−3)2+42r=25r=5\begin{array}{rcl} r&=&\sqrt{(-3)^2+4^2}\\\\ r&=&\sqrt{25}\\\\ r&=&5 \end{array}
Find the trigonometric ratios:

cos⁡θ=−35sin⁡θ=45tan⁡θ=−43\begin{array}{rcl} \cos{\theta}&=&\displaystyle-\frac{3}{5}\\\\ \sin{\theta}&=&\displaystyle\frac{4}{5}\\\\ \tan{\theta}&=&\displaystyle-\frac{4}{3} \end{array}


2. P(12,−32)P\Bigg(\displaystyle\frac{1}{2},\displaystyle-\frac{\sqrt{3}}{2}\Bigg)

Find radius r:

r=(12)2+(−32)2r=14+34r=1r=1\begin{array}{rcl} r&=&\sqrt{\Bigg(\displaystyle\frac{1}{2}\Bigg)^2+\Bigg(\displaystyle-\frac{\sqrt{3}}{2}\Bigg)^2}\\\\ r&=&\sqrt{\displaystyle\frac{1}{4}+\displaystyle\frac{3}{4}}\\\\ r&=&\sqrt{1}\\\\ r&=&1 \end{array}
Find the trigonometric ratios:

cos⁡θ=12sin⁡θ=−32tan⁡θ=(−32)(12)=−3\begin{array}{rcl} \cos{\theta}&=&\displaystyle\frac{1}{2}\\\\ \sin{\theta}&=&\displaystyle-\frac{\sqrt{3}}{2}\\\\ \tan{\theta}&=&\displaystyle\frac{\Bigg(\displaystyle-\frac{\sqrt{3}}{2}\Bigg)}{\Bigg(\displaystyle\frac{1}{2}\Bigg)}&=&-\sqrt{3} \end{array}

Practice: Trigonometric Ratios in Radians


Fill in the blanks.
If the initial angle is in radians, answer in radians. If the initial angle is in degrees, answer in degrees.
Do NOT include units in your answer.

Reference AngleCoterminal Angle between−2π≤θ≤2π; −360∘≤θ≤360∘145∘a.b.11π6c.d.−340∘e.f.−5π3g.h.\begin{array}{|c|l|l|}\hline &\textbf{Reference Angle}&\textbf{Coterminal Angle between}\\ &&\footnotesize-2\pi\leq{}\theta{}\leq{}2\pi;~-360^\circ\leq{}\theta{}\leq{}360^\circ\\\hline\\ 145^\circ&\textbf{a.}&\textbf{b.}\\\\\hline\\ \displaystyle\frac{11\pi}{6}&\textbf{c.}&\textbf{d.}\\\\\hline\\ -340^\circ&\textbf{e.}&\textbf{f.}\\\\\hline\\ -\displaystyle\frac{5\pi}{3}&\textbf{g.}&\textbf{h.}\\\\\hline \end{array}
a.
b.
c.
d.
e.
f.
g.
h.

Practice: Trigonometric Functions of Angles in Standard Position,

Reference & Coterminal Angles


Let P(−x,−x)P(-\sqrt{x},-\sqrt{x}) be a point on the terminal arm of some angle θ.\theta. Find cos⁡θ, sin⁡θ, tan⁡θ\cos{\theta},~\sin{\theta},~\tan{\theta}. Leave answer in exact form.


Practice: Trigonometric Functions of Angles in Standard Position

Reference & Coterminal Angles

Let sin⁡θ<0\sin{\theta}<0 and tan⁡θ>0\tan{\theta}>0.

If cos⁡θ=−53\cos{\theta}=\displaystyle-\frac{\sqrt{5}}{3}, determine sin⁡2θcos⁡θ−tan⁡θ\sin^2{\theta}\cos{\theta}-\tan{\theta}.
Extra Practice
Special Angles & Unit Circle
Let P(−2,3)P(-2,3) be on the terminal arm for some angle θ\theta.
Determine the following trigonometric ratios:
Trigonometric Functions of Angles in Standard Position, Reference & Coterminal Angles
Practice: Trigonometric Functions of Angles in Standard Position
Reference & Coterminal Angles
Let cos⁡θ<0\cos{\theta}<0 and sin⁡θ<0\sin{\theta}<0.
Trigonometric Functions of Angles in Standard Position, Reference & Coterminal Angles
Practice: Trigonometric Functions of Angles in Standard Position
Reference & Coterminal Angles
Let cos⁡θ<0\cos\theta<0 and tan⁡θ<0\tan{\theta}<0.
Trigonometric Functions of Angles in Standard Position, Reference & Coterminal Angles
Practice: Trigonometric Functions of Angles in Standard Position,
Reference & Coterminal Angles
Let P(3x,−5x)P(\sqrt{3x},-\sqrt{5x}) be a point on the terminal arm of some angle θ.\theta. Find cos⁡θ, sin⁡θ, tan⁡θ\cos{\theta},~\sin{\theta},~\tan{\theta}. Leave answer in exact form.
Trigonometric Functions of Angles in Standard Position, Reference & Coterminal Angles
Practice: Trigonometric Functions of Angles in Standard Position,
Reference & Coterminal Angles
Let P(−x3,−x3)P(-\sqrt{x^3},-\sqrt{x^3}) be a point on the terminal arm of some angle θ.\theta. Find cos⁡θ, sin⁡θ, tan⁡θ\cos{\theta},~\sin{\theta},~\tan{\theta}. Leave answer in exact form.
Trigonometric Ratios in Radians
Practice: Trigonometric Ratios in Radians
Fill in the blanks.
If the initial angle is in radians, answer in radians. If the initial angle is in degrees, answer in degrees.
Trigonometric Ratios in Radians
Practice: Trigonometric Ratios in Radians
Fill in the blanks.
If the initial angle is in radians, answer in radians. If the initial angle is in degrees, answer in degrees.