M151Q Precalculus · Unit 3: Quadratics, polynomials, triangles · on Exam 3
TRIG 2.1–2.3 Trigonometric ratios and solving right triangles
Study notes for Trigonometric ratios and solving right triangles 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 Mathub.
Big ideas
- For an acute angle $\theta$ in a right triangle: $\sin\theta = \dfrac{\text{opp}}{\text{hyp}}$, $\cos\theta = \dfrac{\text{adj}}{\text{hyp}}$, $\tan\theta = \dfrac{\text{opp}}{\text{adj}}$ (SOH-CAH-TOA). The ratios depend only on the angle, not the triangle's size (similarity).
- Exact values: $\sin30^\circ = \tfrac12$, $\cos30^\circ = \tfrac{\sqrt3}2$, $\tan30^\circ = \tfrac{1}{\sqrt3}$; $\sin45^\circ = \cos45^\circ = \tfrac{\sqrt2}2$, $\tan45^\circ = 1$; $\sin60^\circ = \tfrac{\sqrt3}2$, $\cos60^\circ = \tfrac12$, $\tan60^\circ = \sqrt3$.
- Solve a triangle = find all sides and angles. From one side and one acute angle: the other angle is $90^\circ - \theta$; use the ratio that links the known side to the wanted side. From two sides: find the third by Pythagoras and an angle by an inverse ratio ($\theta = \tan^{-1}(\text{opp}/\text{adj})$).
- Applications: angle of elevation/depression, ladders, ramps. Draw, label the right angle, mark which side is opposite the known angle.
- Cofunctions: $\sin\theta = \cos(90^\circ - \theta)$.
Key formulas
Ratios
$$\sin\theta = \frac{\text{opp}}{\text{hyp}},\quad \cos\theta = \frac{\text{adj}}{\text{hyp}},\quad \tan\theta = \frac{\text{opp}}{\text{adj}} = \frac{\sin\theta}{\cos\theta}$$
Inverse ratios
$$\theta = \sin^{-1}\frac{\text{opp}}{\text{hyp}},\quad \theta = \cos^{-1}\frac{\text{adj}}{\text{hyp}},\quad \theta = \tan^{-1}\frac{\text{opp}}{\text{adj}}$$
Worked example
A 6 m ladder leans against a wall making $68^\circ$ with the ground. How high does it reach, and how far is its base from the wall?
Show the solution
Height $= 6\sin68^\circ \approx 5.56$ m; base $= 6\cos68^\circ \approx 2.25$ m. Check: $5.56^2 + 2.25^2 \approx 36$ ✓.
Common mistakes
- Using the wrong side as "opposite" (opposite is across from the angle, never the hypotenuse).
- Calculator in radian mode when the angle is in degrees.
- Rounding intermediate values, then failing the Pythagorean check.
Exam tipLabel the triangle O, A, H relative to the angle you are using before choosing a ratio. Relabel if you switch angles.
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