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Wize High School Algebra II Textbook (Common Core) > Trigonometric Ratios (Radians)

Special Angles & Unit Circle in Radians

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Special Angles & Unit Circle in Radians

The unit circle is a special case of a general circle and can be defined as:
x2+y2=1x^2+y^2=1

Since the radius is 1, the trigonometric ratios become:

cos⁡θ=xr=xsin⁡θ=yr=ytan⁡θ=yx\begin{array}{rcl} \cos{\theta}&=&\displaystyle\frac{x}{r}=x\\\\ \sin{\theta}&=&\displaystyle\frac{y}{r}=y\\\\ \tan{\theta}&=&\displaystyle\frac{y}{x}\\\\ \end{array}

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The Unit Circle and Special Triangles

There are 2 special triangles.

π4−π4−π2 △\displaystyle\frac{\pi}{4}-\displaystyle\frac{\pi}{4}-\displaystyle\frac{\pi}{2}~\bigtriangleup and π6−π3−π2 △\displaystyle\frac{\pi}{6}-\displaystyle\frac{\pi}{3}-\displaystyle\frac{\pi}{2}~\bigtriangleup


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The special triangles are taken from the unit circle.

Each special angle on the unit circle has a matching set of (x,y)(x,y) coordinate points.

The C.A.S.T system describes where the trigonometric functions are positive.
  • Cosine is positive in quadrant I & IV.
  • All trigonometric functions are positive in quadrant I.
  • Sine is positive in quadrant I & II.
  • Tangent is positive in quadrant I & III.

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Example

Let's evaluate sin⁡(3π4)\sin{\Big(\displaystyle\frac{3\pi}{4}\Big)} using the unit circle.


Since 3π4 \displaystyle\frac{3\pi}{4}~ is in quadrant II, then sin⁡θ>0\sin{\theta}>0.

So, sin⁡(3π4)=22\sin{\Big(\displaystyle\frac{3\pi}{4}\Big)}=\displaystyle\frac{\sqrt{2}}{2}.

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Reciprocal Trigonometric Ratios: Special Angles & Unit Circle


The reciprocal trigonometric ratios are defined below.
sec⁡θ=rxcsc⁡θ=rycot⁡θ=xy\begin{array}{rcl} \sec{\theta}&=&\displaystyle\frac{r}{x}\\\\ \csc{\theta}&=&\displaystyle\frac{r}{y}\\\\ \cot{\theta}&=&\displaystyle\frac{x}{y}\\\\ \end{array}
The C.A.S.T system also describes where the reciprocal trigonometric functions are positive.
  • Cosine is positive in quadrant I & IV →\color{red}\rightarrow Secant is positive in quadrant I & IV.
  • All trigonometric functions are positive in quadrant I →\color{red}\rightarrow All reciprocal trigonometric functions are positive in quadrant I.
  • Sine is positive in quadrant I & II →\color{red}\rightarrow Cosecant is positive in quadrant I & II.
  • Tangent is positive in quadrant I & III →\color{red}\rightarrow Cotangent is positive in quadrant I & III.

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Example

Let's evaluate csc⁡(3π4)\csc{\Big(\displaystyle\frac{3\pi}{4}\Big)}.

First,

sin⁡(3π4)=22\sin{\Big(\displaystyle\frac{3\pi}{4}\Big)}=\displaystyle\frac{\sqrt{2}}{2}


Therefore, csc⁡θ=22=2\csc{\theta}=\displaystyle\frac{2}{\sqrt{2}}=\sqrt{2}

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Example: Trigonometric Ratios, the Unit Circle, & Special Angles

Evaluate the following:
  1. cos⁡(20π6)  &  sec⁡(20π6)\cos{\Big(\displaystyle\frac{20\pi}{6}\Big)}~~\&~~\sec{\Big(\displaystyle\frac{20\pi}{6}\Big)}
  2. sin⁡(13π3)  &  csc⁡(13π3)\sin{\Big(\displaystyle\frac{13\pi}{3}\Big)}~~\&~~\csc{\Big(\displaystyle\frac{13\pi}{3}\Big)}
  3. tan⁡(13π4)  &  cot⁡(13π4)\tan{\Big(\displaystyle\frac{13\pi}{4}\Big)}~~\&~~\cot{\Big(\displaystyle\frac{13\pi}{4}\Big)}

a. cos⁡(20π6)  &  sec⁡(20π6)\cos{\Big(\displaystyle\frac{20\pi}{6}\Big)}~~\&~~\sec{\Big(\displaystyle\frac{20\pi}{6}\Big)}

The angle 20π6\displaystyle\frac{20\pi}{6} is in quadrant IlI.

So, both cos⁡θ  &  sec⁡θ\cos{\theta}~~\&~~\sec{\theta} will be negative.

Thus,

cos⁡(20π6)=−12sec⁡(20π6)=−2\begin{array}{rcl} \cos{\Big(\displaystyle\frac{20\pi}{6}\Big)}&=&\displaystyle-\frac{1}{2}\\\\ \sec{\Big(\displaystyle\frac{20\pi}{6}\Big)}&=&\displaystyle-2 \end{array}


b. sin⁡(13π3)  &  csc⁡(13π3)\sin{\Big(\displaystyle\frac{13\pi}{3}\Big)}~~\&~~\csc{\Big(\displaystyle\frac{13\pi}{3}\Big)}

The angle 13π3\displaystyle\frac{13\pi}{3} is in quadrant I.

So, both sin⁡θ  &  csc⁡θ\sin{\theta}~~\&~~\csc{\theta} will be positive.

Thus,

sin⁡(13π3)=32csc⁡(13π3)=23\begin{array}{rcl} \sin{\Big(\displaystyle\frac{13\pi}{3}\Big)}&=&\displaystyle\frac{\sqrt{3}}{2}\\\\ \csc{\Big(\displaystyle\frac{13\pi}{3}\Big)}&=&\displaystyle\frac{2}{\sqrt{3}} \end{array}


c. tan⁡(13π4)  &  cot⁡(13π4)\tan{\Big(\displaystyle\frac{13\pi}{4}\Big)}~~\&~~\cot{\Big(\displaystyle\frac{13\pi}{4}\Big)}

The angle 13π4\displaystyle\frac{13\pi}{4} is in quadrant III.

So, both tan⁡θ  &  cot⁡θ\tan{\theta}~~\&~~\cot{\theta} will be positive.

Thus,

tan⁡(13π4)=1cot⁡(13π4)=1\begin{array}{rcl} \tan{\Big(\displaystyle\frac{13\pi}{4}\Big)}&=&1\\\\ \cot{\Big(\displaystyle\frac{13\pi}{4}\Big)}&=&1 \end{array}

Practice: Trigonometric Ratios, the Unit Circle, & Special Angles

Let θ=4π3\theta{}=\displaystyle\frac{4\pi}{3}. Find:
  1. cos⁡θ\cos{\theta}
  2. sin⁡θ\sin{\theta}
  3. tan⁡θ\tan{\theta}
  4. sec⁡θ\sec{\theta}
  5. csc⁡θ\csc{\theta}
  6. cot⁡θ\cot{\theta}
Leave answers in exact form, and rationalize all denominators.

Practice: Trigonometric Ratios, the Unit Circle, & Special Angles

The terminal arm of some angle θ\theta has an endpoint of P(2,−5)P(\sqrt{2},-5). It intersects the unit circle at some point Q(x,y)Q(x, y). What are the coordinate points of Q?Q?

Practice: Trigonometric Ratios, the Unit Circle, & Special Angles

Let tan⁡θ=54  &  sin⁡θ>0\tan{\theta}=\displaystyle\frac{5}{4}~~\&~~\sin{\theta}>0 for some θ\theta in the domain 0≤θ<2π0\leq{}\theta<2\pi. Determine the 5 other trigonometric ratios. Leave the answer in exact form.




Extra Practice
Trigonometric Ratios, the Unit Circle, & Special Angles
Practice: Trigonometric Ratios, the Unit Circle, & Special Angles
Let sin⁡θ=47  &  cos⁡θ>0\sin{\theta}=\displaystyle\frac{4}{7}~~\&~~\cos{\theta}>0 for some θ\theta in the domain 0≤θ<2π0\leq{}\theta<2\pi. Determine the 5 other trigonometric ratios. Leave the answer in exact form.
Trigonometric Ratios, the Unit Circle, & Special Angles
Practice: Trigonometric Ratios, the Unit Circle, & Special Angles
The terminal arm of some angle θ\theta has an endpoint of P(2,−3)P(2,-3). It intersects the unit circle at some point Q(x,y)Q(x, y). What are the coordinate points of Q?Q?
Trigonometric Ratios, the Unit Circle, & Special Angles
Practice: Trigonometric Ratios, the Unit Circle, & Special Angles
Let cos⁡θ=13  &  sin⁡θ<0\cos{\theta}=\displaystyle\frac{1}{3}~~\&~~\sin{\theta}<0 for some θ\theta in the domain 0≤θ<2π0\leq{}\theta<2\pi. Determine the 5 other trigonometric ratios. Leave the answer in exact form.
Trigonometric Ratios, the Unit Circle, & Special Angles
Practice: Trigonometric Ratios, the Unit Circle, & Special Angles
The terminal arm of some angle θ\theta has an endpoint of P(−3,4)P(-\sqrt{3},4). It intersects the unit circle at some point Q(x,y)Q(x, y). What are the coordinate points of Q?Q?
Trigonometric Ratios, the Unit Circle, & Special Angles
Practice: Trigonometric Ratios, the Unit Circle, & Special Angles
Let θ=7π4\theta{}=\dfrac{7\pi}{4}. Find:
cos⁡θ\cos{\theta}
Trigonometric Ratios, the Unit Circle, & Special Angles
Practice: Trigonometric Ratios, the Unit Circle, & Special Angles
Let θ=7π6\theta{}=\dfrac{7\pi}{6}. Find:
cos⁡θ\cos{\theta}