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Learn Index · Trigonometry Grade 9–10

Sine, cosine, and tangent ratios

sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.

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The three ratios

For an acute angle θ in a right triangle:

Ratio Formula Memory
Sine sin θ = opposite / hypotenuse SOH
Cosine cos θ = adjacent / hypotenuse CAH
Tangent tan θ = opposite / adjacent TOA

SOH CAH TOA is the classic way to remember them.

Example

A right triangle has legs 5 (opposite θ) and 12 (adjacent), hypotenuse 13.

  • sin θ = 5/13 ≈ 0.385
  • cos θ = 12/13 ≈ 0.923
  • tan θ = 5/12 ≈ 0.417

Ranges of values

  • Since the hypotenuse is the longest side, sin θ and cos θ are always between 0 and 1 for acute angles.
  • tan θ can be any positive number. tan 45° = 1; tan 89° ≈ 57.

Exact values from special triangles

Two special triangles give exact values.

45°–45°–90° (sides 1, 1, √2)

30°–60°–90° (sides 1, √3, 2; the 1 is opposite 30°)

θ sin θ cos θ tan θ
30° 1/2 √3/2 1/√3
45° √2/2 √2/2 1
60° √3/2 1/2 √3

Relationships between the ratios

  • tan θ = sin θ / cos θ
  • sin θ = cos(90° − θ): the sine of one acute angle equals the cosine of the other. (That’s where "co-sine" comes from: the sine of the complementary angle.)
  • sin²θ + cos²θ = 1 (from the Pythagorean theorem)

Using the calculator

In degree mode:

  • sin 40° ≈ 0.6428
  • cos 40° ≈ 0.7660
  • tan 40° ≈ 0.8391

Setting up equations

  1. Label the sides from the given angle.
  2. Decide which two sides are involved (one known, one unknown).
  3. Choose the ratio that uses those two sides.
  4. Solve.

Example: θ = 28°, adjacent = 15. Find the hypotenuse.

  • Sides: adjacent and hypotenuse → cosine
  • cos 28° = 15/h → h = 15/cos 28° ≈ 16.99

Worked examples

Example 1: Opposite 8, hypotenuse 17. sin θ? 8/17.

Example 2: tan 60° exactly? √3.

Example 3: θ = 52°, opposite 20. Adjacent? tan 52° = 20/a → a = 20/tan 52° ≈ 15.63.

Common mistakes

  • Choosing the wrong ratio because sides weren’t labelled first.
  • Thinking sin θ can be greater than 1.
  • Writing sin as a number without θ: sin means "sine of an angle".

Try it yourself

  1. Opposite 3, adjacent 4. Find all three ratios.
  2. Exact value of cos 30°?
  3. If sin θ = 0.6, what is cos(90° − θ)?
  4. θ = 65°, hypotenuse 12. Find the adjacent side.

Answers: 1) sin 3/5, cos 4/5, tan 3/4 2) √3/2 3) 0.6 4) ≈ 5.07

Quick recap

  • Sine, cosine and tangent are trigonometric ratios of a right triangle’s sides.
  • Remember SOH CAH TOA.
  • Special triangles give exact values for 30°, 45° and 60°.