Right triangle trigonometry connects the angles of a right triangle to the ratios of its side lengths. The memory aid SOHCAHTOA helps students remember which sides go with sine, cosine, and tangent. These ratios are useful because they let you find missing distances and angles without measuring them directly.
They appear in geometry, physics, engineering, navigation, architecture, and many real-world measurement problems.
For any chosen acute angle θ in a right triangle, the hypotenuse is always the side across from the right angle, the opposite side is across from θ, and the adjacent side touches θ but is not the hypotenuse. The trig ratios are sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. To solve for a missing side, choose the ratio that includes the known side and the unknown side, then rearrange the equation.
To solve for a missing angle, use inverse trig functions such as sin^-1, cos^-1, or tan^-1.
Key Facts
- SOHCAHTOA means sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent.
- The hypotenuse is always the longest side and is always opposite the 90° angle.
- For a chosen angle θ, the opposite side is across from θ and the adjacent side touches θ.
- If sin θ = opposite/hypotenuse, then opposite = hypotenuse sin θ and hypotenuse = opposite/sin θ.
- If cos θ = adjacent/hypotenuse, then adjacent = hypotenuse cos θ and hypotenuse = adjacent/cos θ.
- If tan θ = opposite/adjacent, then opposite = adjacent tan θ and adjacent = opposite/tan θ.
Vocabulary
- Right triangle
- A triangle with one angle equal to 90°.
- Hypotenuse
- The side opposite the right angle, and the longest side of a right triangle.
- Opposite side
- The side across from the chosen acute angle θ.
- Adjacent side
- The side next to the chosen acute angle θ that is not the hypotenuse.
- Inverse trigonometric function
- A function such as sin^-1, cos^-1, or tan^-1 that finds an angle from a trig ratio.
Common Mistakes to Avoid
- Labeling opposite and adjacent before choosing the angle θ. These side names depend on the angle you are using, so they can change if you choose the other acute angle.
- Using the hypotenuse as the adjacent side. The adjacent side touches θ, but the hypotenuse is never called adjacent in SOHCAHTOA.
- Choosing a trig ratio that does not contain the known and unknown sides. Pick sine, cosine, or tangent based on the two side lengths involved in the problem.
- Forgetting to use inverse trig when solving for an angle. If the ratio is known and θ is unknown, use sin^-1, cos^-1, or tan^-1 instead of multiplying by a trig value.
Practice Questions
- 1 A right triangle has angle θ = 35° and hypotenuse 12 cm. Find the side opposite θ to the nearest tenth.
- 2 A ladder makes a 70° angle with the ground and reaches 4.8 m up a wall. How long is the ladder to the nearest tenth of a meter?
- 3 In a right triangle, one acute angle is θ. Explain how the labels opposite and adjacent change if you use the other acute angle instead.