Right-angled triangle trigonometry: sine

Maths · Scaling the 1-unit triangle

A triangle with hypotenuse 1 cm and angle 20° has opposite side sin 20° = 0.342 cm (from a table of trig values). Drag the hypotenuse out to 5.6 cm. The two triangles are similar, so every side is multiplied by the same scale factor, and that factor is the new hypotenuse. This bar shows the lengths as numbers, so picture the triangle growing as you drag.

What you need to know

• Label the sides from the marked angle θ: the hypotenuse faces the right angle, the opposite faces θ, the adjacent touches both.
• With a hypotenuse of 1, the opposite side is sin θ: the height where the angle meets the unit circle.
• Sine does not grow in step with the angle: 30° gives a height of 0.5, but 60° only about 0.87.
• Have a goPriya says: "Double the angle, double the sine, so sin 60° must be exactly double sin 30°." What number is she predicting, and is she right?

  She predicts 1. The real height is only about 0.87, so no.

  Sine does not scale linearly with the angle, so doubling 30° to 60° does not double the height of 0.5.

• Any right-angled triangle is similar to the hypotenuse-1 triangle with the same angle, scaled up by its hypotenuse h.
• Have a goA triangle with hypotenuse 1 has an opposite side of 0.4. Scale it up so its hypotenuse is 10. How long is the opposite side now?

  4

  Every side is multiplied by the same scale factor, and the scale factor is the new hypotenuse, 10: 0.4 × 10 = 4.

• So the opposite side is h × sin θ: multiply the hypotenuse by the sine to find the opposite.
• Rearranged, h = opposite ÷ sin θ and sin θ = opposite ÷ hypotenuse. Pick the version with what you want as its subject.
• A table lists sines for only some angles, so reading an angle from it can give just a range.
• A calculator finds the sine of any angle, as long as it is set to degrees (D on the display, not R).
• To find an angle precisely, use sin⁻¹: θ = sin⁻¹(opposite ÷ hypotenuse). For example, sin⁻¹(0.5) = 30°.
• Have a goYou have worked out sin θ = 0.8 and want the angle itself. Do you press sin or sin⁻¹?

  sin⁻¹: it turns the ratio 0.8 back into an angle.

  sin takes an angle and gives a ratio, so to go from the ratio back to the angle you need the inverse.

• Finish by checking a hypotenuse: it is the longest side, so it must be longer than the opposite.

The big picture

Sine is a height: in a triangle with hypotenuse 1, the opposite side is sin θ. Scale that triangle by any hypotenuse h and you get opposite = h × sin θ, which you can rearrange to find a hypotenuse or, with sin⁻¹, an angle.

⚠ Watch out

Two classics. First, thinking sine grows in step with the angle, so sin 60° is double sin 30°; it is 0.5 and about 0.87. Second, multiplying by sin θ to find a hypotenuse, which gives one shorter than the opposite.

Memory hook

The hypotenuse is the big side, and sin θ shrinks it down to the opposite. So going down, multiply by sin θ; climbing back up, divide.

Check yourself

Cover the page. Can you say why sin 60° is not double sin 30°, and whether you multiply or divide by sin θ to find a hypotenuse?