Right Triangle Trigonometry Math Example 2

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Example 2

medium
A ladder 13 feet long leans against a wall. The base of the ladder is 5 feet from the wall. Find the angle the ladder makes with the ground (to the nearest degree).

Solution

  1. 1
    Step 1: Draw the right triangle. The ladder is the hypotenuse (13 ft), the distance from the wall to the base is the adjacent side (5 ft), and the wall is the opposite side.
  2. 2
    Step 2: Use the cosine ratio: cos⁡θ=adjhyp=513\cos\theta = \frac{\text{adj}}{\text{hyp}} = \frac{5}{13}.
  3. 3
    Step 3: Solve for θ\theta: θ=cos⁡−1 ⁣(513)\theta = \cos^{-1}\!\left(\frac{5}{13}\right).
  4. 4
    Step 4: Calculate: 513≈0.3846\frac{5}{13} \approx 0.3846, so θ=cos⁡−1(0.3846)≈67.4°≈67°\theta = \cos^{-1}(0.3846) \approx 67.4° \approx 67°.

Answer

The ladder makes approximately 67°67° with the ground.
When you know two sides of a right triangle and want an angle, use the inverse trig functions. Here, the adjacent side and hypotenuse are known, so cos⁡−1\cos^{-1} is appropriate. As a check: sin⁡θ=1213\sin\theta = \frac{12}{13} (opposite side found via Pythagoras: 132−52=12\sqrt{13^2-5^2}=12), giving θ≈67.4°\theta \approx 67.4° — consistent.

About Right Triangle Trigonometry

The three primary trigonometric ratios—sine, cosine, and tangent—defined as ratios of specific sides in a right triangle.

Learn more about Right Triangle Trigonometry →

More Right Triangle Trigonometry Examples

Example 1 easy

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Example 3 easy

In a right triangle with angle [formula] and hypotenuse [formula], find both legs.

Example 4 hard

From the top of a 50-meter tall lighthouse, the angle of depression to a boat is [formula]. How far

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