Trigonometric Functions
Grade 11 · Math · Study Guide
Explore the six trig functions, their graphs, key identities, and radian measure on the unit circle.
On a right triangle, sin(theta) = opposite/hypotenuse, cos(theta) = adjacent/hypotenuse, tan(theta) = opposite/adjacent. On the unit circle, the point (x, y) at angle theta gives x = cos(theta) and y = sin(theta).
Radian measure links angle to arc length: 180 degrees = pi radians. Key angles to memorize: 0, pi/6, pi/4, pi/3, pi/2 and their quadrant equivalents.
The Pythagorean identity sin^2(theta) + cos^2(theta) = 1 is foundational. Dividing by cos^2(theta) gives 1 + tan^2(theta) = sec^2(theta); dividing by sin^2(theta) gives cot^2(theta) + 1 = csc^2(theta).
| Angle | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 0 |
| 30 | pi/6 | 1/2 | sqrt(3)/2 | sqrt(3)/3 |
| 45 | pi/4 | sqrt(2)/2 | sqrt(2)/2 | 1 |
| 60 | pi/3 | sqrt(3)/2 | 1/2 | sqrt(3) |
| 90 | pi/2 | 1 | 0 | undefined |
| Function | Definition |
|---|---|
| csc(theta) | 1/sin(theta) |
| sec(theta) | 1/cos(theta) |
| cot(theta) | cos(theta)/sin(theta) |
Key terms
- Unit circle
- A circle of radius 1 centered at the origin; any point on it is (cos(theta), sin(theta)).
- Radian
- An angle measure where 2*pi radians = 360 degrees; one radian is subtended by an arc equal to the radius.
- Pythagorean identity
- sin^2(theta) + cos^2(theta) = 1; true for all angles.
- Period
- The horizontal length of one complete cycle; sin and cos have period 2*pi, tan has period pi.
Practice questions
In a right triangle, sin(theta) = 3/5. What is cos(theta) (assume theta is acute)?
- 4/5
- 3/4
- 5/3
- sqrt(34)/5
Show answer
sin^2 + cos^2 = 1: (9/25) + cos^2 = 1 -> cos^2 = 16/25 -> cos = 4/5.
Convert 210 degrees to radians.
- pi/6
- 7pi/4
- 7pi/6
- 5pi/6
Show answer
210 * (pi/180) = 210pi/180 = 7pi/6.
What are the unit circle coordinates at theta = pi/3?
Show answer
cos(pi/3) = 1/2, sin(pi/3) = sqrt(3)/2. Point: (1/2, sqrt(3)/2).
What is the period of y = sin(2x)?
- pi
- 2*pi
- 4*pi
- pi/2
Show answer
Period = 2*pi / |b| = 2*pi / 2 = pi.
If cos(theta) = 5/13, what is |sin(theta)|?
- 12/13
- 8/13
- 13/5
- 5/12
Show answer
sin^2 = 1 - (25/169) = 144/169. |sin| = 12/13.
Fun facts
- Sound and light are both modeled by sine waves; the pitch of a musical note corresponds to its sine wave frequency in Hz.
- The word sine traces its roots through Arabic translation of a Sanskrit word from ancient Indian astronomy used to describe half-chords.
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