10-Minute Trig
Copyright © 2002–2023 by Stan Brown, BrownMath.com
Copyright © 2002–2023 by Stan Brown, BrownMath.com
Summary: Use this page to jog your memory with the basic facts of right-angle trigonometry. Please see Trig without Tears for explanations of these quick notes and many more topics.
circumference of circle: 2πr
angle around circle (like clock hand): 2π radians
or 360°
Therefore
2π = 360°, or
π = 180°, or
1 radian = 180°/π.
Press [MODE]
to check calculator mode (radian
or degree).
SOHCAHTOA | sin A = cos(π/2−A)
cos A = sin(π/2−A) tan A = cot(π/2−A) cot A = tan(π/2−A) sec A = csc(π/2−A) csc A = sec(π/2−A) |
cot A = 1 / tan A
sec A = 1 / cos A csc A = 1 / sin A |
The definitions based on an acute angle in a right triangle extend to trig functions of any angle:
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sin θ = y/r
cos θ = x/r tan θ = sin θ / cos θ = y/x |
cot θ = 1 / tan θ
sec θ = 1 / cos θ csc θ = 1 / sin θ |
Pythagorean theorem ( y² + x² = r² ) leads to
sin² θ + cos² θ = 1>
r is always >0, so signs of functions in any quadrant pop right out from signs of x and y in that quadrant. Do quadrant angles by reference to x y r, e.g. cos 0° = 1/1 = 1 and sin π = 0/−1 = 0. Use reference angle (acute angle between terminal side and x axis) to relate function values to values for an acute angle. |
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c = 1 (given), a = b
By Pythagoras, a = b = √2 / 2 |
c = 1 (given), b = c/2 = ½
By Pythagoras, a = √3 / 2 |
0 = 0° | π/6 = 30° | π/4 = 45° | π/3 = 60° | π/2 = 90° | |
---|---|---|---|---|---|
sin θ | 0 | 1/2 | √2 / 2 | √3 / 2 | 1 |
cos θ | 1 | √3 / 2 | √2 / 2 | 1/2 | 0 |
tan θ | 0 | √3 / 3 | 1 | √3 | undef. |
arcsin 0.65 or sin-10.65 means the angle
whose sine is 0.65. That’s not the same as 1/sin 0.65
Function ranges:
−π/2 ≤ arcsin x ≤ +π/2,
0 ≤ arccos x ≤ +π,
−π/2 < arctan x < +π/2
Function composition (see diagram at right):
What is e.g. cos( arctan x )?
Solution: arctan x is the angle whose tangent is
x;
call it A. Then you must find cos A.
Use Pythagoras to find the third side, √x²+1, then read off
function value: cos A =
1 / √x²+1.
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