﻿ Trigonometric Values of Special Angles

# Trigonometric Values of Special Angles

With the work done in prior sections (particularly those listed below), we have everything needed to efficiently find exact values for things like $\,\cos\frac{81\pi}{4}\,$ and $\,\csc (-2640^\circ)\,.$

All the necessary tools/ideas are repeated below, in-a-nutshell. Having trouble following the brief discussion on this page? If so, review the links above (in order)—they offer a much slower and kinder approach.

## SOHCAHTOA

Sine  Opposite  Hypotenuse

## Reciprocal Relationships for Trig Functions

For example:  $\displaystyle\csc = \frac{1}{\sin}$

## Trigonometric Values of Small Special Angles

 angle/number sine, $\,\sin = \frac{\text{OPP}}{\text{HYP}}$ cosine, $\,\cos = \frac{\text{ADJ}}{\text{HYP}}$ tangent, $\,\tan = \frac{\text{OPP}}{\text{ADJ}}$ cotangent(reciprocal of tangent) secant(reciprocal of cosine) cosecant(reciprocal of sine)
 angle/number $0^\circ = 0 \text{ rad}$ sine $0$ cosine $1$ tangent $0$ cotangent not defined secant $1$ cosecant not defined
 angle/number $\displaystyle 30^\circ = \frac{\pi}{6} \text{ rad}$ sine $\displaystyle\frac 12$ cosine $\displaystyle\frac{\sqrt 3}2$ tangent $\displaystyle\frac1{\sqrt 3} = \frac{\sqrt 3}{3}$ cotangent $\sqrt 3$ secant $\displaystyle\frac{2}{\sqrt 3} = \frac{2\sqrt 3}{3}$ cosecant $2$
 angle/number $\displaystyle 45^\circ = \frac{\pi}{4} \text{ rad}$ sine $\displaystyle\frac 1{\sqrt 2} = \frac{\sqrt 2}{2}$ cosine $\displaystyle\frac 1{\sqrt 2} = \frac{\sqrt 2}{2}$ tangent $1$ cotangent $1$ secant $\sqrt 2$ cosecant $\sqrt 2$
 angle/number $\displaystyle 60^\circ = \frac{\pi}{3} \text{ rad}$ sine $\displaystyle\frac{\sqrt 3}{2}$ cosine $\displaystyle \frac{1}{2}$ tangent $\displaystyle \sqrt 3$ cotangent $\displaystyle\frac1{\sqrt 3} = \frac{\sqrt 3}{3}$ secant $2$ cosecant $\displaystyle\frac{2}{\sqrt 3} = \frac{2\sqrt 3}{3}$
 angle/number $\displaystyle 90^\circ = \frac{\pi}{2} \text{ rad}$ sine $1$ cosine $0$ tangent not defined cotangent $0$($\cot := \frac{\cos}{\sin}$) secant not defined cosecant $1$

## Signs ($\pm$) of Sine, Cosine, and Tangent in All Quadrants

Reciprocals retain the sign ($+/-$) of the original number. Therefore, in all the quadrants:

• Cosecant has the same sign as sine.
• Secant has the same sign as cosine.
• Cotangent has the same sign as tangent.

## Trigonometric Values for Arbitrary Special Angles

In Special Triangles and Common Trigonometric Values, the ‘Locate-Shrink/Size-Signs’ method was introduced for finding trigonometric values of special angles. With additional tools and terminology now at hand, that discussion is presented more generally and efficiently here.

## The RRQSS (Reduce-Reference/Quadrant-Size/Sign) Technique

### Reduce:

[This step is optional. If your angle isn't too big (say, $\,510^\circ\,$ or $\,-\frac{7\pi}{3}\,$), then it may be easy for you to find its reference angle and quadrant, without ‘reducing’ it first. Your choice!]

As discussed in Reference Angles, remove any extra rotations from $\,\theta\,$:

 $\theta\,$ in DEGREES How many extra rotations (if any) in $\,\theta\,$? To answer this question, compute $\,n := \frac{|\theta|}{360^\circ}\,,$ rounded to the nearest whole number. If $\,\theta\,$ is positive, then replace $\,\theta\,$ by: $$\theta - n\cdot 360^\circ$$ If $\,\theta\,$ is negative, then replace $\,\theta\,$ by: $$\theta + n\cdot 360^\circ$$ Now, your angle/number is manageable: it is between $\,-180^\circ\,$ and $\,180^\circ\,.$ But, it has the same terminal point, so all the trigonometric values are the same!
 $\theta\,$ in RADIANS How many extra rotations (if any) in $\,\theta\,$? To answer this question, compute $\,n := \frac{|\theta|}{2\pi}\,,$ rounded to the nearest whole number. If $\,\theta\,$ is positive, then replace $\,\theta\,$ by: $$\theta - n\cdot 2\pi$$ If $\,\theta\,$ is negative, then replace $\,\theta\,$ by: $$\theta + n\cdot 2\pi$$ Now, your angle/number is manageable: it is between $\,-\pi\,$ and $\,\pi\,.$ But, it has the same terminal point, so all the trigonometric values are the same!

Lay off $\,\theta\,$ in the standard way:

• Start at the positive $x$-axis
• Positive angles are swept out in a counterclockwise direction; start by going up
• Negative angles are swept out in a clockwise direction; start by going down

Determine the reference angle/number for $\,\theta\,.$

Determine the quadrant for $\,\theta\,.$

### Size/Sign:

Use the reference angle/number to find the correct size of the desired trigonometric value.

Use the quadrant to find the correct sign of the desired trigonometric value.

## Examples

In this first example, the angles aren't too big, so the optional (reduce) step is skipped.

 Find:  $\,\sec 510^\circ\,$ Reduce (skipped) [Dr. Burns would work with $\,510^\circ - 360^\circ = 150^\circ\,.$] Reference/Quadrant $\,510^\circ\,$ is in quadrant II;  the reference angle is $\,30^\circ\,$ Size/Sign Size:  $\displaystyle\sec 30^\circ = \frac{2}{\sqrt 3}\,$ Sign:  In quadrant II, the secant is negative. Thus: $$\sec 510^\circ = -\frac{2}{\sqrt 3}$$
 Find:  $\displaystyle\,\cot (-\frac{7\pi}{3})\,$ Reduce (skipped) [Dr. Burns would work with $\,-\frac{7\pi}3 + \frac{6\pi}3 = -\frac{\pi}3\,.$] Reference/Quadrant $\displaystyle\,-\frac{7\pi}{3}\,$ is in quadrant IV;  the reference angle is $\,\displaystyle\frac{\pi}{3}$ Size/Sign Size:  $\displaystyle\cot\frac{\pi}{3} = \frac{1}{\sqrt 3}\,$ Sign:  In quadrant IV, the cotangent is negative. Thus: $$\cot(-\frac{7\pi}{3}) = -\frac{1}{\sqrt 3}$$

In this final example, the angles are very big, so get rid of extra rotations (reduce) in the first step:

 Find:  $\,\csc(-2640^\circ)$ Reduce $$\cssId{s134}{\frac{|\theta|}{360^\circ} = \frac{2640^\circ}{360^\circ} \approx 7}$$ $$\cssId{s135}{-2640^\circ + 7\cdot 360^\circ = -120^\circ}$$ Work with $\,-120^\circ\,$ instead of $\,-2640^\circ\,.$ Reference/Quadrant $\,-120^\circ\,$ is in quadrant III;  the reference angle is $\,60^\circ\,$ Size/Sign SIZE:  $\displaystyle\csc 60^\circ = \frac{2}{\sqrt 3}\,$ SIGN:  In quadrant III, the cosecant is negative. Thus: $$\csc(-2640^\circ) = -\frac{2}{\sqrt 3}$$
 Find:  $\displaystyle\,\cos (\frac{81\pi}{4})\,$ Reduce $$\cssId{s142}{\frac{|\theta|}{2\pi} = \frac{81\pi/4}{2\pi} \approx 10}$$ $$\cssId{s143}{\frac{81\pi}{4} - 10\cdot 2\pi = \frac{81\pi}{4} - \frac{80\pi}{4} = \frac{\pi}4}$$ Work with $\displaystyle\,\frac{\pi}4\,$ instead of $\displaystyle\,\frac{81\pi}4\,.$ Reference/Quadrant $\,\frac{\pi}4\,$ is in quadrant I; the reference angle is $\,\displaystyle\frac{\pi}{4}$ Size/Sign SIZE:  $\displaystyle\cos\frac{\pi}{4} = \frac{1}{\sqrt 2}$ SIGN:  In quadrant I, the cosine is positive. Thus: $$\cos\frac{81\pi}{4} = \frac{1}{\sqrt 2}$$