Sine, cosine, and tangent
Trigonometry links angles to side ratios in right triangles. With it, one angle and one side are enough to find the rest of a right triangle.
The three ratios
For an acute angle \(\theta\) in a right triangle: \(\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}\), and \(\tan\theta = \frac{\text{opposite}}{\text{adjacent}}\). The memory aid SOH-CAH-TOA helps. Because similar triangles have equal ratios, these values depend only on the angle, not the triangle's size.
Reading the triangle
Opposite and adjacent depend on which angle you use. In a 3-4-5 triangle, for the angle opposite the side of 3: \(\sin\theta = 0.6\), \(\cos\theta = 0.8\), and \(\tan\theta = 0.75\). For the other acute angle, sine and cosine swap.
Exact values
From the special triangles: \(\sin 30° = \tfrac{1}{2}\), \(\cos 30° = \tfrac{\sqrt{3}}{2}\), \(\tan 30° = \tfrac{1}{\sqrt{3}}\); \(\sin 45° = \cos 45° = \tfrac{\sqrt{2}}{2}\), \(\tan 45° = 1\); \(\sin 60° = \tfrac{\sqrt{3}}{2}\), \(\cos 60° = \tfrac{1}{2}\), \(\tan 60° = \sqrt{3}\).
Finding sides
If a ramp makes a 10° angle with the ground and is 6 m long, its height is \(6\sin 10° \approx 6 \times 0.174 \approx 1.04\) m. Choose the ratio that links the side you know with the side you want; make sure your calculator is in degree mode.
Finding angles
Inverse trigonometric functions recover angles. If a right triangle has opposite side 5 and adjacent side 12, then \(\tan\theta = \tfrac{5}{12}\), so \(\theta = \tan^{-1}(\tfrac{5}{12}) \approx 22.6°\). The notation \(\tan^{-1}\) means inverse tangent, not 1 over tangent.
Angles of elevation and depression
An angle of elevation is measured upward from the horizontal, such as looking up at a kite; an angle of depression is measured downward, such as looking down from a cliff to a boat. Because horizontal lines are parallel, the angle of depression from A to B equals the angle of elevation from B to A.
A key identity
Since \(\text{opposite}^2 + \text{adjacent}^2 = \text{hypotenuse}^2\), dividing by the hypotenuse squared gives \(\sin^2\theta + \cos^2\theta = 1\) for every angle. Also \(\tan\theta = \frac{\sin\theta}{\cos\theta}\), and the sine of an angle equals the cosine of its complement.
Worked example
From 50 m away, the angle of elevation to the top of a building is 32°. Its height above eye level is \(50\tan 32° \approx 50 \times 0.625 \approx 31.2\) m. If the observer's eyes are 1.6 m above the ground, the building is about 32.8 m tall.
Key takeaways
In a right triangle, sine is opposite over hypotenuse, cosine adjacent over hypotenuse, and tangent opposite over adjacent (SOH-CAH-TOA); the ratios depend only on the angle. Opposite and adjacent depend on which angle you use. Special triangles give exact values for 30, 45, and 60 degrees. Use the ratio linking the known and unknown sides, and inverse functions to find angles. Angles of elevation and depression between two points are equal. sin squared + cos squared = 1, and sine of an angle equals cosine of its complement.
Flashcard
Opposite / hypotenuse
Check your understanding
In a 3-4-5 right triangle, what is the sine of the angle opposite the side of length 3?