Sine, cosine, tangent & right triangles
← Back to Math TopicsStuck on the arithmetic? Use this mini calculator for square roots, trig values and inverse trig.
In a right triangle, the three main trig ratios connect an angle to the sides. Remember them with SOH-CAH-TOA:
The names of the sides depend on angle θ.
The longest side
Always opposite the right angle (90°).
Across from angle θ
The side that does not touch the angle you are using.
Next to angle θ
The side touching θ that is not the hypotenuse.
\(a^2 + b^2 = c^2\)
c is the hypotenuse. Find a missing side without angles.
Use the inverse functions on your calculator:
\(\theta = \sin^{-1}\!\left(\dfrac{\text{opp}}{\text{hyp}}\right)\)
\(\theta = \cos^{-1}\!\left(\dfrac{\text{adj}}{\text{hyp}}\right)\)
\(\theta = \tan^{-1}\!\left(\dfrac{\text{opp}}{\text{adj}}\right)\)
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 0.5 | 0.866 | 0.577 |
| 45° | 0.707 | 0.707 | 1 |
| 60° | 0.866 | 0.5 | 1.732 |
| 90° | 1 | 0 | undefined |
A right triangle has an angle of 30° and a hypotenuse of 10. Find the side opposite the 30° angle.
\(\cot\theta = \dfrac{1}{\tan\theta}\)
\(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\)
\(\cot\theta = \dfrac{\cos\theta}{\sin\theta}\)
\(\sin^2\theta + \cos^2\theta = 1\)
Here \(\sin^2\theta\) means \((\sin\theta)^2\).
\(1 + \tan^2\theta = \dfrac{1}{\cos^2\theta}\)
\(1 + \cot^2\theta = \dfrac{1}{\sin^2\theta}\)
\(\sin 2\theta = 2\sin\theta\cos\theta\)
\(\cos 2\theta = \cos^2\theta - \sin^2\theta\)
\(\cos 2\theta = 2\cos^2\theta - 1\)
\(\cos 2\theta = 1 - 2\sin^2\theta\)
\(\tan 2\theta = \dfrac{2\tan\theta}{1 - \tan^2\theta}\)
\(\sin\dfrac{\theta}{2} = \pm\sqrt{\dfrac{1 - \cos\theta}{2}}\)
\(\cos\dfrac{\theta}{2} = \pm\sqrt{\dfrac{1 + \cos\theta}{2}}\)
\(\tan\dfrac{\theta}{2} = \dfrac{1 - \cos\theta}{\sin\theta}\)
\(\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B\)
\(\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B\)
\(\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}\)
These work for all triangles, not just right triangles. Sides \(a, b, c\) are opposite angles \(A, B, C\).
\(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\)
\(c^2 = a^2 + b^2 - 2ab\cos C\)
\(\cos(-\theta) = \cos\theta\)
\(\sin(-\theta) = -\sin\theta\)
\(\tan(-\theta) = -\tan\theta\)
\(\sin(90^\circ - \theta) = \cos\theta\)
\(\cos(90^\circ - \theta) = \sin\theta\)
If \(\sin\theta = 0.6\) and \(\cos\theta = 0.8\), find \(\sin 2\theta\).