M151Q Precalculus · Unit 4: Trigonometric functions · on Final Exam
TRIG 8.2–8.3 Inverse trigonometric functions and the reciprocal functions
Study notes for Inverse trigonometric functions and the reciprocal functions in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in StudyCat.
Big ideas
- Sine, cosine and tangent are not one-to-one, so their inverses use restricted domains: $\sin^{-1}x$ returns angles in $[-\pi/2, \pi/2]$, $\cos^{-1}x$ in $[0,\pi]$, $\tan^{-1}x$ in $(-\pi/2, \pi/2)$. Inputs to $\sin^{-1}$ and $\cos^{-1}$ must lie in $[-1,1]$.
- Consequence: $\sin^{-1}(\sin\theta) = \theta$ only when $\theta$ is in the restricted range. $\sin^{-1}(\sin 150^\circ) = 30^\circ$.
- Solving $\sin\theta = k$ over a full circle needs the inverse value and its partner ($180^\circ - $ value for sine, $360^\circ -$ value for cosine, $+180^\circ$ for tangent).
- Reciprocal functions: $\sec\theta = 1/\cos\theta$, $\csc\theta = 1/\sin\theta$, $\cot\theta = 1/\tan\theta$; undefined where the denominator is 0. Identities $1 + \tan^2\theta = \sec^2\theta$, $1 + \cot^2\theta = \csc^2\theta$.
- Exact evaluations: $\cos(\sin^{-1}\tfrac35)$: draw the right triangle with opposite 3, hypotenuse 5, adjacent 4, so $\tfrac45$.
Key formulas
Restricted ranges
$$\sin^{-1}: [-\tfrac\pi2, \tfrac\pi2],\qquad \cos^{-1}: [0, \pi],\qquad \tan^{-1}: (-\tfrac\pi2, \tfrac\pi2)$$
Reciprocals
$$\sec\theta = \frac{1}{\cos\theta},\quad \csc\theta = \frac1{\sin\theta},\quad \cot\theta = \frac{1}{\tan\theta}$$
Worked example
Evaluate $\cos^{-1}(-\tfrac{\sqrt2}{2})$, $\sin^{-1}(\sin 200^\circ)$, and $\tan(\cos^{-1}\tfrac{5}{13})$.
Show the solution
$\cos^{-1}(-\tfrac{\sqrt2}2) = 135^\circ$ (QII, in $[0,180^\circ]$). $\sin 200^\circ = -\sin 20^\circ$, so $\sin^{-1}$ gives $-20^\circ$. Triangle with adjacent 5, hypotenuse 13, opposite 12: $\tan = \tfrac{12}{5}$.
Common mistakes
- Returning an angle outside the restricted range.
- Writing sin⁻¹x as 1/sin x (that is csc x).
Exam tipFor sin⁻¹ and tan⁻¹ think "between −90° and 90°"; for cos⁻¹ think "between 0° and 180°". Then pick the quadrant from the sign of the input.
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