Grade 11 · Free math lesson

Trigonometry

After this lesson, you will be able to find missing sides and angles in a right triangle using SOH-CAH-TOA.

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Skills in this unit

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Standards covered:G-SRT.C.6G-SRT.C.8F-TF.A.1F-TF.A.2F-TF.B.5F-IF.C.7.eG-SRT.D.10G-SRT.D.11F-TF.C.8

Label Your Triangle

Every right triangle has three special sides named based on the angle you pick. The side across from your angle is Opposite, the side next to it is Adjacent, and the longest side is always the Hypotenuse.

Example: Pick the 30° angle in a right triangle. The side directly across from it is Opposite. The side touching it (not the longest) is Adjacent. The longest side is the Hypotenuse.

Remember: HYPOtenuse = HYPO = the BIG one, always across from the right angle!

Meet SOH-CAH-TOA

SOH-CAH-TOA is a word trick that tells you three ratios: Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent. Pick the one that uses the sides you already know!

Example: If your angle is 40°, you know the Opposite side (5) and want the Hypotenuse — use SOH: sin(40°) = 5 ÷ Hypotenuse.

Remember: Say it out loud: Some Old Hippo Caught A Hippopotamus Trying Our Apples — SOH-CAH-TOA!

Solve for the Missing Side

Write your ratio as a fraction, then cross-multiply or rearrange to find the unknown side. Use your calculator to get the exact decimal value of sin, cos, or tan.

Example: sin(30°) = Opposite ÷ 10. So Opposite = 10 × sin(30°) = 10 × 0.5 = 5. The missing side is 5!

Remember: Multiply when the unknown is on TOP, divide when it's on the BOTTOM of the fraction.

Worked example

A ramp makes a 25° angle with the ground. The ramp is 12 m long. How high does it rise?

  1. 1Step 1 — Label: The ramp is the Hypotenuse (12 m). The height is Opposite to the 25° angle.
  2. 2Step 2 — Pick ratio: We have Opposite and Hypotenuse, so use SOH → sin(25°) = Opposite ÷ 12.
  3. 3Step 3 — Solve: Opposite = 12 × sin(25°) = 12 × 0.4226 ≈ 5.07 m.

Answer: The ramp rises about 5.07 metres.

The #1 mistake to avoid

Students mix up Opposite and Adjacent because they forget to check WHICH angle they picked first — always circle your angle before labelling the sides!

Key takeaways

  • Label sides as Opposite, Adjacent, and Hypotenuse based on your chosen angle.
  • Use SOH-CAH-TOA to pick the right ratio for the sides you have.
  • Rearrange the equation and use your calculator to find the missing side or angle.

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