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Solving Right Triangles
Pythagorean Theorem & Trigonometry

Master the tools for finding missing sides and missing angles in right triangles using \( a^2 + b^2 = c^2 \) and the sine, cosine, and tangent ratios.

Geometry

What Is a Right Triangle?

A B C \(a\) (leg) \(b\) (leg) \(c\) (hypotenuse) \(\alpha\) \(\beta\) \(90^\circ\)

Key Vocabulary

Hypotenuse
The side opposite the right angle: always the longest side of the triangle.
Opposite Leg
The leg across from a given acute angle: the side it "looks at."
Adjacent Leg
The leg that touches the given acute angle (not the hypotenuse).

Why Solving Right Triangles Matters

Right triangles appear everywhere: architecture, navigation, physics, engineering, and computer graphics. Learning to find every missing measurement gives you a universal problem-solving tool.

Finding Sides
Use the Pythagorean Theorem when you know two sides.
Finding Angles
Use inverse trig functions when you know two sides and need an angle.
Mixed Problems
Use trig ratios when you know one side and one acute angle.

The Pythagorean Theorem

\(a\) \(b\) \(c\) \(a^2\) \(b^2\) \(a^2 + b^2 = c^2\)

The Trigonometric Ratios: SOH-CAH-TOA

\(\theta\) Adjacent Opposite Hypotenuse SOH \(\sin\theta = \frac{\text{Opp}}{\text{Hyp}}\) CAH \(\cos\theta = \frac{\text{Adj}}{\text{Hyp}}\) TOA \(\tan\theta = \frac{\text{Opp}}{\text{Adj}}\)

Sine & Cosine of Complementary Angles

\(\alpha\) \(\beta\) \(a\) \(b\) \(c\) \(\sin\alpha = \frac{b}{c}\) \(\cos\beta = \frac{b}{c}\) They're equal!

Pythagorean vs. Trig Ratios: When to Use Which

Pythagorean Theorem Trig Ratios (SOH-CAH-TOA)
You know Two sides One side + one acute angle
You find The third side A missing side or angle
Formula \( a^2 + b^2 = c^2 \) \( \sin, \cos, \tan \) ratios
Angles needed? No Yes (at least one acute angle)

How to Solve a Right Triangle

1
Sketch & label. Draw the triangle, mark the right angle, and label every known side and angle.
2
Identify what's missing. Decide whether you need a side or an angle.
3
Choose your tool. Two sides known: use \( a^2 + b^2 = c^2 \). One side + one angle: use SOH-CAH-TOA. Two sides, need an angle: use inverse trig.
4
Solve & check. Compute the answer, then verify: do all three angles sum to \( 180^\circ \)? Does the hypotenuse stay the longest side?

Example 1: Finding the Hypotenuse

6 8 \(c = ?\) A B C

Example 2: Finding a Missing Leg

\(a = ?\) \(b = 12\) \(c = 13\)

Example 3: Using Trig to Find a Side

\(35^\circ\) adjacent \(x = ?\) 20 (opposite) (hypotenuse)

Example 4: Finding a Missing Angle

\(\theta = ?\) 7 9 (adjacent) (opposite)

Example 5: Complementary Angles in Action

\(33^\circ\) \(57^\circ\) P Q R \(a\) \(b\) \(c\)

Common Mistakes to Avoid

✕ Using \( a^2 + b^2 = c^2 \) Wrong
Incorrectly setting up the equation, for example as \(c^2 + a^2 = b^2\). The hypotenuse (\(c\)) must always be by itself on one side of the equation.
✕ Swapping Opposite and Adjacent
Forgetting that opposite/adjacent depend on which angle you're using. Always label relative to your chosen angle.
✕ Calculator in Wrong Mode
Using radians when the problem is in degrees: always check your calculator is set to DEG mode.

Summary: Your Right Triangle Toolkit

Tool Formula Use When
Pythagorean Theorem \( a^2 + b^2 = c^2 \) Know 2 sides, find the 3rd
Sine \( \sin\theta = \frac{\text{opp}}{\text{hyp}} \) Have angle + hyp or opp
Cosine \( \cos\theta = \frac{\text{adj}}{\text{hyp}} \) Have angle + hyp or adj
Tangent \( \tan\theta = \frac{\text{opp}}{\text{adj}} \) Have angle + opp or adj
Complementary Rule \( \sin\theta = \cos(90^\circ - \theta) \) Relate sin & cos of complementary angles

Question 1 of 5

A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?

Question 2 of 5

In a right triangle, the hypotenuse is 10 and one leg is 6. What is the other leg?

Question 3 of 5

In a right triangle with an acute angle of \(40^\circ\) and a hypotenuse of 15, which expression gives the length of the side opposite the \(40^\circ\) angle?

Question 4 of 5

If \( \sin 25^\circ \approx 0.4226 \), what is the value of \( \cos 65^\circ \)?

Question 5 of 5

A right triangle has legs of length 8 and 15. What is the measure of the angle opposite the side of length 8, to the nearest degree?