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 $\,\csc (-2640^\circ)\,.$
- Special Triangles and Common Trigonometric Values
- Reference Angles
- Radian Measure
- the Trigonometric Functions
- Signs of All the Trigonometric Functions
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.
Two Special Triangles
SOHCAHTOA
Sine
Opposite
Hypotenuse
Cosine
Adjacent
Hypotenuse
Tangent
Opposite
Adjacent
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 |
|
$\theta\,$ in RADIANS |
|
Reference/Quadrant:
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.
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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: $$\cos\frac{81\pi}{4} = \frac{1}{\sqrt 2}$$ |
Concept Practice
- Choose a specific problem type, or click ‘New problem’ for a random question.
- Think about your answer.
- Click ‘Check your answer’ to check!