Trigonometric functions and their properties

( \pi+\theta) is in the third quadrant where sine is negative. In exams identifying allied angles is important.

( \pi+\theta) is in the third quadrant and cosine is negative. In exams recall the sign table quickly.

The period of ( \tan \theta) is ( \pi), so ( \tan(\pi+\theta)=\tan \theta). In exams remember the periodicity of tangent.

(2\pi-\theta) is in the fourth quadrant where tangent is negative. In exams derive tangent from sine and cosine signs.

( \sin \theta) is zero at points like (0), (\pi), and (2\pi). In exams write the general solution with (n\in\mathbb{Z}).

( \cos \theta) is zero at odd multiples of ( \frac{\pi}{2}). In exams remember the pattern ( \frac{(2n+1)\pi}{2}).

( \tan \theta=0) occurs when ( \sin \theta=0) and ( \cos \theta\neq0). In exams use period ( \pi) for tangent.

The combined solution of ( \sin \theta=\sin \alpha) is ( \theta=n\pi+(-1)^n\alpha). In exams learn to write two cases in one formula.

For ( \cos \theta=\cos \alpha), angles occur in both directions. In exams do not forget ( \pm\alpha).

The period of ( \tan \theta) is ( \pi), so the solution is ( \theta=n\pi+\alpha). In exams use period ( \pi) for tangent equations.

At \( \frac{\pi}{2}-\theta\), tangent changes to its co-function cotangent. In exams remember complementary angle property.

At \( \frac{\pi}{2}-\theta\), cotangent changes to the co-function tangent. In exams remember reciprocal function pairs.

From ( \sin \theta=\cos \theta), ( \tan \theta=1), and in the first quadrant ( \theta=\frac{\pi}{4}). In exams divide by ( \cos \theta) only when it is nonzero.

The reference angle for ( \sin \theta=\frac{1}{2}) is ( \frac{\pi}{6}), and sine is positive in the first and second quadrants. In exams choose only answers in the given interval.

For ( \cos \theta=\frac{1}{2}), the reference angle is ( \frac{\pi}{3}), and cosine is positive in the first and fourth quadrants. In exams write both solutions up to (2\pi).

The reference angle for ( \tan \theta=\sqrt{3}) is ( \frac{\pi}{3}), and in (0<\theta<\pi) tangent is positive in the first quadrant. In exams choose the quadrant by sign.

Because (1-\cos^2 \theta=\sin^2 \theta), the ratio is (1). In exams always check the denominator condition.

Because (1-\sin^2 \theta=\cos^2 \theta), the value is (1). In exams replace the numerator using the basic identity.

In ( \tan \theta=\frac{4}{3}), opposite is (4), adjacent is (3), and hypotenuse is (5). In exams identify Pythagorean triplets.

In ( \cot \theta=\frac{12}{5}), adjacent is (12), opposite is (5), and hypotenuse is (13). In exams use ( \cos \theta=\frac{\text{adjacent}}{\text{hypotenuse}}).