Trigonometric Functions
What you must understand
Trigonometric functions connect angles to ratios. In a right triangle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent; the reciprocals are cosecant, secant, and cotangent. The unit circle extends these definitions to all angles, with the standard reference angles (30, 45, 60 degrees and their radian equivalents) worth memorizing in each quadrant. Key identities include the Pythagorean identity sin^2 + cos^2 = 1, the reciprocal identities, and the quotient identity tan = sin/cos. Radian measure relates to degrees by pi radians equals 180 degrees. Graphs of sine and cosine have amplitude and period 2pi, while tangent has period pi. You should evaluate exact values, convert units, and solve simple trigonometric equations within a stated interval.
Traps to watch
- Using degree mode when the problem is in radians, or vice versa.
- Forgetting the sign of a function in a given quadrant (All Students Take Calculus).
- Reversing opposite and adjacent when the triangle is oriented differently.
- Ignoring that trigonometric equations usually have multiple solutions in an interval.
- Confusing amplitude with period.
Worked example
Find sin(theta) and cos(theta) for theta = 120 degrees. The reference angle is 60 degrees, in quadrant II where sine is positive and cosine is negative. So sin(120) = sqrt(3)/2 and cos(120) = -1/2.
Rule of thumb
Reduce to the reference angle, then apply the quadrant sign; check whether the problem expects degrees or radians.