Sine, Cosine, and Tangent: Solving Right-Triangle Problems With Ladders, Ramps, and Slopes

Worked Example: The Ladder

A 10-foot ladder leans against a wall at an angle of 75° from the ground. How high does it reach, and how far is the base from the wall? The ladder is the hypotenuse. The height is the side opposite the 75° angle: height = 10 × sin 75° = 10 × 0.9659 = 9.66 feet. The distance from the wall is the adjacent side: 10 × cos 75° = 10 × 0.2588 = 2.59 feet.

A common safety guideline is the 4-to-1 rule: for every 4 feet of height, place the base 1 foot from the wall, which corresponds to an angle of about 76°. That is close to the 75° used in this example.

Finding an Angle: Ramps

A wheelchair ramp must rise 30 inches over a run of 30 feet (360 inches), following the common accessibility guideline of a 1:12 slope. The angle is arctan(30 ÷ 360) = arctan(0.0833) = 4.76°. The ramp surface length is √(30² + 360²) = 361.2 inches, about 30.1 feet. Use the inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) to find angles from side ratios.

A 1:12 slope means 8.33% grade, because grade is rise divided by run expressed as a percentage. Roads and trails are often described by percent grade: a 10% grade is an angle of arctan(0.10) = 5.71°.

Heights Using Tangent

You can measure the height of a tree or building without climbing it. Stand 50 feet from the base of a tree and measure the angle of elevation to the top as 35°. The height above your eye level is 50 × tan 35° = 50 × 0.7002 = 35.0 feet. Add your eye height, say 5.5 feet, for a total of 40.5 feet. Shadows work too: if a 6-foot person has a shadow 4 feet long, the angle of the sun is arctan(6 ÷ 4) = 56.3°, and a tree with a 30-foot shadow is 45 feet tall.

Clinometer apps on a phone can measure the angle accurately enough for rough estimates.

The Calculator Mode Mistake

Calculators work in degrees or radians. If your calculator is set to radians and you enter sin 75, you get -0.3878, which is wrong for a 75° angle. Check the mode indicator before every trigonometry session. A quick test: sin 30 should return 0.5 in degree mode.

Another mistake is mixing up which side is opposite and adjacent. Always identify the angle first, then the side across from it (opposite), the side next to it that is not the hypotenuse (adjacent), and the longest side (hypotenuse).

Beyond Right Triangles

For triangles without a right angle, use the law of sines (a ÷ sin A = b ÷ sin B) and the law of cosines (c² = a² + b² − 2ab cos C). Roof pitch, surveying, and navigation often use these. The Pythagorean theorem is a special case of the law of cosines for a 90° angle.