Practice Maths

Introduction to Trigonometry

Key Ideas

Key Terms

hypotenuse
Always the longest side — opposite the right angle.
opposite
Side is directly across from the reference angle θ.
adjacent
Side is next to the reference angle θ (but not the hypotenuse).
SOH
Sin θ = Opposite ÷ Hypotenuse
CAH
Cos θ = Adjacent ÷ Hypotenuse
TOA
Tan θ = Opposite ÷ Adjacent

Naming the sides relative to angle θ

In a right triangle, once you choose the reference angle θ (never the right angle), you can label all three sides. The side opposite the right angle is always the hypotenuse. The side directly facing θ is the opposite. The remaining side (between θ and the right angle) is the adjacent.

Hot Tip The hypotenuse never changes — it is always the side opposite the right angle. But the opposite and adjacent sides do swap when you change which acute angle you call θ. Always re-label the triangle each time you change your reference angle.

Worked Example

Question: A right triangle has a 35° angle at vertex A, a right angle at vertex B, and a 55° angle at vertex C. The side from A to C (hypotenuse) = 10 cm, the side from B to C = 8.19 cm, and the side from A to B = 5.74 cm. Label the sides relative to the 35° angle, then write sin(35°) as a fraction.

Step 1 — Identify the hypotenuse.
The right angle is at B, so the side opposite B is AC. Therefore AC = 10 cm is the hypotenuse.

Step 2 — Identify the opposite side (relative to 35°).
The side directly across from 35° at A is BC = 8.19 cm → this is the opposite side.

Step 3 — Identify the adjacent side.
The remaining side AB = 5.74 cm lies between 35° and the right angle → this is the adjacent side.

Step 4 — Write sin(35°) using SOH.
sin(35°) = opposite ÷ hypotenuse = BC ÷ AC = 8.19 ÷ 10 = 0.574 (approximately)

What is Trigonometry?

Trigonometry is the branch of mathematics that connects angles to side lengths in right-angled triangles. The word comes from Greek: trigonon (triangle) + metron (measure). It is used everywhere — from building ramps and bridges to GPS navigation and game design.

The key insight is this: for any given angle in a right-angled triangle, the ratio of two specific sides is always the same, no matter how big or small the triangle is. That ratio is what the trig functions measure.

Naming the Three Sides

Before you can use trigonometry, you must correctly label the three sides of a right-angled triangle relative to your chosen reference angle (called θ):

  • Hypotenuse (H): The longest side, always opposite the right angle. It never changes regardless of which angle you choose.
  • Opposite (O): The side directly across from the reference angle θ. It changes depending on which angle you pick.
  • Adjacent (A): The side next to the reference angle θ (but not the hypotenuse). It also changes depending on which angle you pick.

Practice: draw a right triangle, mark a corner angle as θ, then label O, A, and H. Now imagine moving θ to a different corner — notice that O and A swap roles.

SOHCAHTOA — The Three Trig Ratios

Three ratios link angles to sides. The mnemonic SOH CAH TOA helps you remember them:

  • SOH: sin(θ) = Opposite ÷ Hypotenuse   (O/H)
  • CAH: cos(θ) = Adjacent ÷ Hypotenuse   (A/H)
  • TOA: tan(θ) = Opposite ÷ Adjacent   (O/A)

These ratios are fixed for any particular angle. For example, sin(30°) = 0.5 in every right-angled triangle that has a 30° angle — big or small, it doesn't matter.

Why Are the Ratios Always the Same?

If two right-angled triangles share the same angle θ, they are similar triangles — their sides scale up or down proportionally. Because the sides scale by the same factor, the ratios of the sides stay identical. This is why sin(30°) is always exactly 1/2, for every right triangle with a 30° angle. The ratios are a property of the angle alone.

How to Label Sides for a Given Angle

Follow this three-step process every time:

  1. Find the right angle (mark it with a small square) — the side opposite it is always the hypotenuse.
  2. Identify your reference angle θ — the side directly across from it is the opposite.
  3. The remaining side, touching θ but not the hypotenuse, is the adjacent.
Common mistake: Students sometimes confuse which angle is θ and mislabel opposite and adjacent. Always start by marking the right angle and the reference angle clearly before labelling any sides. The hypotenuse is ALWAYS opposite the right angle — never the reference angle.

Mastery Practice

  1. For each right triangle described below, name the hypotenuse, opposite, and adjacent side relative to the marked angle θ. Fluency

    1. A right triangle with sides PQ, QR, and PR. The right angle is at Q and θ is at P. Name each side.
    2. A right triangle with sides AB, BC, and AC. The right angle is at B and θ is at A. Name each side.
    3. A right triangle with sides DE, EF, and DF. The right angle is at E and θ is at D. Name each side.
    4. Using the same triangle as (c) but now θ is at F. How do the opposite and adjacent labels change?
    5. A right triangle with sides JK, KL, and JL. The right angle is at K and θ is at J. Which side is the hypotenuse?
    6. In triangle XYZ, the right angle is at Y and θ is at X. Name the opposite and adjacent sides.
    7. Triangle RST has a right angle at S. If θ = angle R, what are the opposite and adjacent sides?
    8. Triangle MNO has a right angle at N and the two acute angles are at M and O. If θ is at O, which side is opposite, which is adjacent, and which is the hypotenuse?

  2. For each situation, state which trigonometric ratio (sin, cos, or tan) connects the two given sides to angle θ. Write the full ratio name. Fluency

    1. You know the opposite side and the hypotenuse.
    2. You know the adjacent side and the hypotenuse.
    3. You know the opposite side and the adjacent side.
    4. You want to find the hypotenuse and you know the opposite side and θ.
    5. You want to find the adjacent side and you know the hypotenuse and θ.
    6. You want to find the opposite side and you know the adjacent side and θ.
    7. The known sides are labelled 7 (adjacent) and 25 (hypotenuse). Which ratio uses these two sides?
    8. The known sides are labelled 9 (opposite) and 40 (adjacent). Which ratio uses these two sides?

  3. Set up the trigonometric ratio as a fraction. Do not evaluate — just write the equation. Fluency

    1. In a right triangle, θ = 42°, opposite = 8, hypotenuse = 12. Write sin(42°) as a fraction.
    2. In a right triangle, θ = 28°, adjacent = 15, hypotenuse = 17. Write cos(28°) as a fraction.
    3. In a right triangle, θ = 55°, opposite = 11, adjacent = 9. Write tan(55°) as a fraction.
    4. A right triangle has θ = 63°, hypotenuse = 20, opposite = x. Write sin(63°) using x.
    5. A right triangle has θ = 37°, hypotenuse = 14, adjacent = x. Write cos(37°) using x.
    6. A right triangle has θ = 48°, adjacent = 6, opposite = x. Write tan(48°) using x.

  4. Choosing the correct ratio and reasoning. Understanding

    1. θ=50° 13 (opp) adj 17 (hyp) A right triangle has θ = 50°, opposite = 13, and hypotenuse = 17. Keanu says he should use cos(50°) = 13/17. Explain why Keanu is wrong and write the correct equation.
    2. In triangle ABC, angle A = 90°, angle B = 32°, AB = 9, BC = 17, AC = 9.01. Priya wants to write a ratio for angle B using sides AB and BC. She writes sin(32°) = 9/17.
      1. Which sides are AB and BC relative to angle B?
      2. Is Priya’s equation correct? Justify your answer.
    3. Write all three trigonometric ratios (sin, cos, tan) for a right triangle where θ = 36°, opposite = 7.3, adjacent = 10.0, hypotenuse = 12.4.
    4. Why can the sine ratio never be greater than 1? Use the definition sin θ = O/H and properties of right triangles to explain.

  5. Match diagrams to equations. Problem Solving

    θ 4 m (opp) 3 m 5 m (hyp)
    Ladder problem
    θ 1.5 m (opp) 6 m (adj)
    Ramp problem
    1. A ladder leans against a wall. The ladder is the hypotenuse (5 m), the wall is the opposite side (4 m), and the ground is the adjacent side (3 m). The angle at the base of the ladder is θ.
      1. Write sin θ, cos θ, and tan θ as fractions.
      2. Which equation would you use to find θ if you only knew the ladder length and the wall height?
    2. A ramp rises 1.5 m over a horizontal distance of 6 m. The angle of the ramp is θ.
      1. Which sides do you know relative to θ? (opposite, adjacent, or hypotenuse?)
      2. Write the trigonometric equation that connects these two known sides with θ.
    3. Three students each write a different equation for the same right triangle with acute angle α, where the side opposite α is 5, the adjacent side is 12, and the hypotenuse is 13.
      • Amelia: sin α = 5/13
      • Ben: cos α = 12/13
      • Cass: tan α = 5/12
      Are all three equations correct? Explain and verify using SOH-CAH-TOA.

  6. Mixed trig ratio challenge. Problem Solving

    Two angles, same triangle. A right triangle ABC has the right angle at B. The hypotenuse AC = 26, AB = 10, and BC = 24. Angle A = α and angle C = γ.
    1. Write sin α, cos α, and tan α as exact fractions.
    2. Write sin γ, cos γ, and tan γ as exact fractions.
    3. Notice that sin α = cos γ and cos α = sin γ. Explain why this is always true for the two acute angles of a right triangle.

  7. Setting up from a description. Problem Solving

    Roof pitch. A roofing contractor describes a roof as having a “pitch angle” of θ at the base, a horizontal run of 6 m (adjacent), and a vertical rise of 4 m (opposite). The rafter length (hypotenuse) is unknown.
    1. Write all three trig ratios for θ using the known measurements.
    2. Which ratio would you use to find the rafter length if you only knew the rise and the pitch angle? Write the equation (do not evaluate).
    3. The contractor states: “I always use tan θ = rise/run because it avoids the unknown hypotenuse.” Explain why this is a practical strategy.

  8. Decide and justify. Problem Solving

    Choosing a ratio strategically. For each scenario below, decide which trig ratio to set up and explain your reasoning. Do not solve — just write the equation.
    1. You are given θ = 58° and the hypotenuse = 15 m. You want to find the side opposite θ.
    2. You are given θ = 31° and the side opposite θ = 9 cm. You want to find the adjacent side.
    3. You are given θ = 44° and the side adjacent to θ = 11 m. You want to find the hypotenuse.
    4. You know all three sides but no angles. Explain which ratio(s) you could use to find an angle, and why there is more than one option.

  9. Misconception correction. Problem Solving

    Find the errors. Three students made mistakes when setting up trig ratios. For each, identify the error and write the correct equation.
    1. Student A, for a triangle with θ = 22°, opposite = 9, hypotenuse = 23, writes: cos(22°) = 9/23.
    2. Student B, for a triangle with θ = 67°, adjacent = 5, opposite = 12, writes: sin(67°) = 12/5.
    3. Student C, for a triangle with θ = 40°, adjacent = 7, hypotenuse = 11, writes: tan(40°) = 7/11.

  10. Real context: trigonometry from a map. Problem Solving

    Navigation problem. A ship leaves port and sails 50 km due east (this is the adjacent side to an angle θ at the port), then turns and sails to a destination point that is 30 km north of the port (the opposite side). The direct route from port to destination is the hypotenuse.
    1. Write sin θ, cos θ, and tan θ using the measurements 30 km (opposite) and 50 km (adjacent).
    2. Use Pythagoras’ theorem to find the exact length of the hypotenuse. Leave your answer in surd form, then evaluate to 2 decimal places.
    3. Rewrite all three trig ratios now that you know the hypotenuse.
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