Section 5.4

Right Triangle Trigonometry

Right triangle trig connects the unit circle to real-world measurement — finding heights, distances, and angles. The cofunction identities also show up in Chapter 7 identity proofs. Word problems from this section are a near-guaranteed topic on Exam 3.

⚡ Quick Summary

Right triangle trig is how math measures the real world — the height of a building, the distance to a ship, the angle of a ramp. The idea is simple: if you know one angle and one side of a right triangle, you can figure out everything else. The magic mnemonic is SOH-CAH-TOA: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent.

SOH-CAH-TOA

\sin\theta = \frac{\text{opp}}{\text{hyp}}, \quad \cos\theta = \frac{\text{adj}}{\text{hyp}}, \quad \tan\theta = \frac{\text{opp}}{\text{adj}}

Cofunction identity (sine–cosine)

\sin\theta = \cos\!\left(\frac{\pi}{2} - \theta\right), \quad \cos\theta = \sin\!\left(\frac{\pi}{2} - \theta\right)

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

The most important step is drawing and labeling the triangle before doing any math. "Opposite" and "adjacent" change depending on which angle you're looking at — the hypotenuse (the longest side, across from the right angle) is the only side that stays fixed. Once you label correctly, just pick the SOH-CAH-TOA ratio that connects what you know to what you need.

There's also a neat relationship called cofunction identities: in a right triangle, the two acute angles add to 90°, so the sine of one angle equals the cosine of the other. That's literally what the "co" in cosine means — "complement's sine." For example, $\sin(30°) = \cos(60°)$ because 30° and 60° are complementary.

For word problems involving angles of elevation (looking up) or angles of depression (looking down), remember that both are measured from the horizontal line, not the vertical. Set up your triangle with the angle at the observer, and the rest follows from SOH-CAH-TOA.

Interactive Diagram

Key Formulas

SOH-CAH-TOA

\sin\theta = \frac{\text{opp}}{\text{hyp}}, \quad \cos\theta = \frac{\text{adj}}{\text{hyp}}, \quad \tan\theta = \frac{\text{opp}}{\text{adj}}

Cofunction identity (sine–cosine)

\sin\theta = \cos\!\left(\frac{\pi}{2} - \theta\right), \quad \cos\theta = \sin\!\left(\frac{\pi}{2} - \theta\right)

Cofunction identity (tangent–cotangent)

\tan\theta = \cot\!\left(\frac{\pi}{2} - \theta\right), \quad \cot\theta = \tan\!\left(\frac{\pi}{2} - \theta\right)

Cofunction identity (secant–cosecant)

\sec\theta = \csc\!\left(\frac{\pi}{2} - \theta\right), \quad \csc\theta = \sec\!\left(\frac{\pi}{2} - \theta\right)

Pythagorean theorem

a^2 + b^2 = c^2

where $c$ is the hypotenuse

Key Takeaways

✓SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent
✓"Opposite" and "adjacent" are relative to the angle you're working with — always label the triangle first
✓Cofunction identities: $\sin\theta = \cos(90° - \theta)$ — the 'co' in cosine literally means 'complement'
✓Angle of elevation (looking up) equals angle of depression (looking down) by alternate interior angles

⚠️ Common Mistakes

✗Labeling 'opposite' and 'adjacent' from the wrong angle — these labels change depending on which angle you're working with; always ask 'opposite to *which* angle?'
✗Using the wrong trig ratio — double-check that the two sides you're connecting match SOH, CAH, or TOA before solving
✗Forgetting to use the inverse trig function when solving for an angle — e.g., writing $\theta = \frac{3}{8}$ instead of $\theta = \arctan\!\left(\frac{3}{8}\right)$
✗Mixing up angle of elevation and angle of depression in word problems — both are measured from the horizontal, not the vertical

Next: Section 6.1 Builds on: 5.2 Builds on: 5.3

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5.3 The Other Trigonometric Functions

6.1 Graphs of Sine & Cosine