Trigonometry
Graphs and exact values
- $\sin$ and $\cos$ wave between $-1$ and $1$ and repeat every $360^\circ$; $\tan$ shoots off at $90^\circ$ and $270^\circ$.
- Learn the exact values:
| $\theta$ | $30^\circ$ | $45^\circ$ | $60^\circ$ |
|---|---|---|---|
| $\sin$ | $\tfrac12$ | $\tfrac{1}{\sqrt2}$ | $\tfrac{\sqrt3}{2}$ |
| $\cos$ | $\tfrac{\sqrt3}{2}$ | $\tfrac{1}{\sqrt2}$ | $\tfrac12$ |
| $\tan$ | $\tfrac{1}{\sqrt3}$ | $1$ | $\sqrt3$ |
Practice
What is the exact value of sin 30°?
sin 30° = 1/2 = 0.5.
Practice
What is the value of tan 45°?
tan 45° = 1.
Identities & solving
- The two key identities: $\tan\theta \equiv \dfrac{\sin\theta}{\cos\theta}$ and $\sin^2\theta + \cos^2\theta \equiv 1$.
- To solve a trig equation: reduce it to one function (often a quadratic in $\sin$ or $\cos$), then find every solution in the interval.
Practice
For any angle θ, what does sin²θ + cos²θ equal?
This is the identity sin²θ + cos²θ ≡ 1.
Practice
Which identity is correct?
tan θ ≡ sin θ / cos θ; and sin²θ + cos²θ ≡ 1.
You've got it
Key idea
- $\sin, \cos$ stay in $[-1, 1]$, repeat every $360^\circ$; know the $30/45/60$ exact values
- identities: $\tan\theta \equiv \dfrac{\sin\theta}{\cos\theta}$, $\sin^2\theta + \cos^2\theta \equiv 1$
- solve by reducing to one trig function, then finding all solutions