Trigonometric Functions and Triangle Solving
Trigonometric functions and triangle solving
Master the unit circle, radian measure, reference angles, and the graphs of sine, cosine, and tangent, including amplitude, period, phase shift, and asymptotes. Know the Pythagorean identities, sum and double-angle formulas, and the transformations used to reduce expressions to a single trigonometric function. For triangles, the sine rule a/sin A = b/sin B = c/sin C, the cosine rule, and the area formula (1/2)ab sin C are essential, together with the fact that angles sum to π.
Traps: losing the second solution when sin θ=k or cos θ=k has two values in [0,2π]; mixing degrees and radians; applying the sine rule in the ambiguous SSA case without checking whether the obtuse alternative leaves a valid angle sum; and using the cosine rule with the wrong included angle. Also watch domain restrictions when solving equations and the sign of trigonometric functions in each quadrant.
Worked example: solve 2 sin x=1 for 0≤x<2π. Then sin x=1/2, so x=π/6 or x=5π/6. Both lie in the interval; the second is easy to miss if you only use the reference angle. For a triangle with a=2, b=√3, A=60°, the sine rule gives sin B=(b sin A)/a=3/4. The acute B is admissible; the obtuse alternative would make A+B exceed 180°, so it is rejected. Rule of thumb: after solving a trigonometric equation or an SSA triangle, test every candidate against the allowed domain or angle sum.