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Trigonometry

11 min read

Sides of a right-angled triangle

Pick one of the two acute angles, . The side opposite the right angle is the hypotenuse. The side facing is the opposite. The side next to (not the hypotenuse) is the adjacent.

As the angle grows from 0° to 90°, the opposite side gets longer and the adjacent side shorter.
Remember SOH CAH TOA.
Sine, cosine and tangentA right-angled triangle with an angle theta at the left. The side opposite theta is the opposite, the side next to theta that is not the longest is the adjacent, and the longest side is the hypotenuse. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse and tangent is opposite over adjacent.θoppositeadjacenthypotenusesin θ = opposite ÷ hypotenusecos θ = adjacent ÷ hypotenusetan θ = opposite ÷ adjacentSOH CAH TOA
Try itChange the angle

Slide the angle in a triangle with hypotenuse 1. Watch how sine, cosine and tangent change. The graph shows sin θ (green) and cos θ (blue).

Special angles
θ30°45°60°
sin θ1/2√2/2 ≈ 0.707√3/2 ≈ 0.866
cos θ√3/2 ≈ 0.866√2/2 ≈ 0.7071/2
tan θ1/√3 ≈ 0.5771√3 ≈ 1.732
Worked exampleFind a side

A right-angled triangle has hypotenuse 10 cm and angle . Find the side opposite .

  1. Opposite and hypotenuse: use sine.
  2. Rearrange.

Answer: The opposite side is 5 cm.

Angles of elevation and depression

The angle of elevation is the angle you look up from the horizontal. The angle of depression is the angle you look down from the horizontal. Both are measured from the horizontal line.

Worked exampleHeight of a tree

You stand 20 m from a tree. The angle of elevation of its top is . Find the height of the tree (to 1 decimal place).

  1. The distance on the ground is adjacent, the height is opposite: use tangent.
  2. Rearrange and use tan 35° ≈ 0.700.

Answer: The tree is about 14.0 m tall (plus the height of your eyes if you want the whole tree).

Check yourself In a right-angled triangle the side next to angle θ and not the hypotenuse is called…
Check yourself Which ratio is opposite ÷ adjacent?

Key terms

Opposite
The side facing the chosen angle.
Adjacent
The side next to the chosen angle that is not the hypotenuse.
Angle of elevation
The angle above the horizontal.
Angle of depression
The angle below the horizontal.

Summary

  • sin = opp/hyp, cos = adj/hyp, tan = opp/adj (SOH CAH TOA).
  • Label the sides from the chosen angle first.
  • Elevation and depression are measured from the horizontal.
Pythagoras Theorem Sets