Solving Right Triangles:
Pythagorean Theorem & Trigonometry
Master the tools to find every missing side and angle in a right triangle: from \( a^2 + b^2 = c^2 \) to sine, cosine, and tangent.
Precalculus
What Is a Right Triangle?
Key Vocabulary
Hypotenuse
The side opposite the right angle: always the longest side, labeled \( c \).
Opposite Side
The leg across from a given acute angle. Changes depending on which angle you reference.
Adjacent Side
The leg that touches a given acute angle (and also touches the right angle).
Why Solving Right Triangles Matters
Every time you know some measurements of a right triangle (a side and an angle, or two sides), you can find all the rest. This skill powers:
Navigation
Pilots and sailors use right-triangle trig to compute headings and distances.
Engineering
Architects calculate roof pitch, ramp slope, and structural loads with these ratios.
Physics
Force vectors, projectile motion, and wave analysis all decompose into right triangles.
The Pythagorean Theorem
Trig Ratios: SOH CAH TOA
SOH
\( \sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}} \)
CAH
\( \cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}} \)
TOA
\( \tan\theta = \dfrac{\text{opposite}}{\text{adjacent}} \)
Sine and Cosine of Complementary Angles
Pythagorean vs. Trigonometric Approach
| Pythagorean Theorem | Trig Ratios | |
|---|---|---|
| What you need | Two sides | One side + one acute angle |
| What you find | The third side | Any missing side or angle |
| Formula | \( a^2 + b^2 = c^2 \) | \(\sin, \cos, \tan\) ratios |
| Limitation | Cannot find angles directly | Requires knowing an angle |
How to Solve a Right Triangle
1
Sketch and label. Draw the triangle, mark the right angle, and label all known sides and angles.
2
Identify what you need. Decide whether you need a missing side or a missing angle.
3
Choose your tool. Two sides known: use \( a^2+b^2=c^2 \). One side + one angle known: use \(\sin, \cos, \tan\).
4
Solve and verify. Compute the unknown. Then check: do all three angles sum to \(180^\circ\)? Does \(a^2+b^2=c^2\)?
Example 1: Find the Hypotenuse
Example 2: Find a Missing Leg
Example 3: Use Trig to Find a Side
Example 4: Use Inverse Trig to Find an Angle
Example 5: Complementary Angle Identity
In a right triangle, \(\angle P = 28^\circ\) and \(\angle Q = 62^\circ\). Verify that \(\sin 28^\circ = \cos 62^\circ\).
1
Note that \(28^\circ + 62^\circ = 90^\circ\), so the angles are complementary.
2
Both \(\sin 28^\circ\) and \(\cos 62^\circ\) reference the same side ratio: opposite to \(P\) over hypotenuse = adjacent to \(Q\) over hypotenuse.
3
Compute: \(\sin 28^\circ \approx 0.4695\) and \(\cos 62^\circ \approx 0.4695\). ✓ They match!
Common Mistakes
✕ Mixing Up Sides
Using a leg as the hypotenuse (or vice versa) in \(a^2+b^2=c^2\). Remember: \(c\) is always the longest side, opposite the \(90^\circ\) angle.
✕ Wrong Trig Ratio
Confusing opposite/adjacent relative to the angle. Always ask: "Which sides does this specific angle touch?"
✕ Calculator in Wrong Mode
Computing \(\sin 30^\circ\) with the calculator set to radians gives \(\approx -0.988\) instead of \(0.5\). Always check your mode.
Summary: Solving Right Triangles
| Given | Tool | Formula |
|---|---|---|
| Two legs | Pythagorean | \( c = \sqrt{a^2+b^2} \) |
| Hypotenuse + one leg | Pythagorean | \( a = \sqrt{c^2-b^2} \) |
| Angle + hypotenuse | Sine or Cosine | \(\text{side} = c \cdot \sin\theta\) or \(c \cdot \cos\theta\) |
| Angle + one leg | Tan (or sin/cos) | \(\text{other leg} = \text{leg} \cdot \tan\theta\) |
| Two sides, need angle | Inverse trig | \(\theta = \sin^{-1}, \cos^{-1}, \) or \(\tan^{-1}\) |
Question 1 of 5
A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?
Apply \( c = \sqrt{a^2+b^2} = \sqrt{25+144} = \sqrt{169} = 13 \). The answer \(17\) comes from adding the legs (wrong operation), and \(169\) forgets the square root.
Question 2 of 5
In a right triangle, the hypotenuse is 15 and one leg is 9. What is the other leg?
Use \( a = \sqrt{c^2 - b^2} = \sqrt{225 - 81} = \sqrt{144} = 12 \). Choice (a) subtracts the values directly, and (b) adds the squares instead of subtracting.
Question 3 of 5
In a right triangle, \(\angle A = 40^\circ\) and the hypotenuse is 18. Which expression gives the length of the side opposite \(\angle A\)?
\(\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}\), so opposite \(= \text{hyp}\times\sin\theta = 18\sin 40^\circ\). Choice (b) gives the adjacent side, and (c) uses tangent which requires a leg, not the hypotenuse.
Question 4 of 5
If \(\sin 25^\circ \approx 0.4226\), what is \(\cos 65^\circ\)?
Since \(25^\circ + 65^\circ = 90^\circ\), the angles are complementary. By the identity \(\sin\theta = \cos(90^\circ-\theta)\), we get \(\cos 65^\circ = \sin 25^\circ \approx 0.4226\). Choice (a) is \(\cos 25^\circ\), not \(\cos 65^\circ\).
Question 5 of 5
A right triangle has legs \( a = 8 \) and \( b = 15 \). What is \(\tan A\) where \(\angle A\) is opposite leg \( a \)?
\(\tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b} = \frac{8}{15}\). Choice (a) is \(\sin A\), choice (b) is \(\cos A\), and choice (c) is \(\tan B\) (the other acute angle's tangent).