Trigonometric Ratios
In a right triangle with angle theta, let P be the perpendicular, B the base and H the hypotenuse. Sine is P over H, cosine is B over H, and tangent is P over B. Cosecant, secant and cotangent are the reciprocals of sine, cosine and tangent. Tangent also equals sine over cosine.
Identities
Sine squared plus cosine squared equals one. One plus tangent squared equals secant squared. One plus cotangent squared equals cosecant squared.
Special Angle Values
Sine of 0, 30, 45, 60 and 90 degrees is 0, one half, one over root 2, root 3 over 2, and 1. Cosine takes the same values in reverse order. Tangent of 0, 30, 45 and 60 degrees is 0, one over root 3, 1 and root 3. Tangent of 90 degrees is undefined.
Complementary Angles
Sine of 90 minus theta equals cosine theta, and tangent of 90 minus theta equals cotangent theta.
Compound and Double Angles
Sine of A plus B is sine A cosine B plus cosine A sine B. Cosine of A plus B is cosine A cosine B minus sine A sine B. Sine of 2A equals 2 sine A cosine A. Cosine of 2A equals cosine squared A minus sine squared A.
Radians
Pi radians equal 180 degrees. Pi over 6 is 30 degrees, pi over 4 is 45, pi over 3 is 60, and pi over 2 is 90.
Heights and Distances
The tangent of the angle of elevation equals the height divided by the horizontal distance.
Shortcuts
Remember sine values as root 0, root 1, root 2, root 3, root 4, each over 2. If sine theta is 3 over 5, use the 3, 4, 5 triple: cosine is 4 over 5 and tangent is 3 over 4. Tangent of 30 times tangent of 60 equals 1.
Worked Examples
Example 1: If sine theta is 3 over 5, find cosine and tangent.
Solution: The base is 4, so cosine is 4 over 5 and tangent is 3 over 4.
Example 2: A 10 metre ladder makes 60 degrees with the ground. How high does it reach?
Solution: 10 times sine 60, which is 5 root 3, about 8.66 metres.
Example 3: Find the height of a tower seen at 30 degrees from 50 metres away.
Solution: 50 times tangent 30, which is about 28.87 metres.
Quick Check
Question 1: What is sine squared plus cosine squared?
Answer: One.
Question 2: What is tangent of 45 degrees?
Answer: One.
Question 3: Convert pi over 6 radians to degrees.
Answer: 30 degrees.