Learning Objectives
  • Identify opposite, adjacent and hypotenuse relative to a chosen acute angle.
  • Use sine, cosine and tangent to find unknown sides.
  • Use inverse trigonometric functions to find acute angles.
  • Solve two-dimensional right-triangle problems.
  • Choose an efficient ratio and round answers appropriately.
Key Terms
Sine
The ratio opposite divided by hypotenuse.
Cosine
The ratio adjacent divided by hypotenuse.
Tangent
The ratio opposite divided by adjacent.
Opposite side
The side directly across from the chosen angle.
Adjacent side
The non-hypotenuse side touching the chosen angle.
Inverse trigonometric function
A calculator operation used to obtain an angle from a ratio.
Angle of elevation
An angle measured upward from a horizontal line.
8th Edition Chapter Map
  • Naming sides relative to an angle
  • Sine, cosine and tangent
  • Finding sides
  • Finding angles
  • Two-dimensional applications
Naming The Sides

The hypotenuse is fixed because it is opposite the right angle. The labels opposite and adjacent depend on the acute angle being used. The adjacent side is the side next to the angle that is not the hypotenuse.

\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}
\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}
\tan\theta=\frac{\text{opposite}}{\text{adjacent}}
Choosing A Ratio

Identify the two sides involved. Use sine for opposite and hypotenuse, cosine for adjacent and hypotenuse, and tangent for opposite and adjacent. A small labelled sketch often prevents errors.

Known / required sides Ratio
Opposite and hypotenuse Sine
Adjacent and hypotenuse Cosine
Opposite and adjacent Tangent
Finding A Side In The Numerator
Worked Example: Find The Opposite Side

Question: A right triangle has hypotenuse 18 cm and an acute angle of 37^\circ. Find the side opposite the angle.

  1. \sin37^\circ=x/18.
  2. x=18\sin37^\circ.
  3. x\approx10.8327.

Answer: 10.8 cm to 3 significant figures.

Finding A Side In The Denominator
Worked Example: Rearrangement

Question: The side adjacent to 52^\circ is 9 m. Find the hypotenuse h.

  1. \cos52^\circ=9/h.
  2. h=9/\cos52^\circ.
  3. h\approx14.618.

Answer: 14.6 m to 3 significant figures.

Finding An Angle

Form the correct ratio, then use the inverse key on the calculator. Make sure the calculator is in degree mode.

Worked Example: Inverse Tangent

Question: In a right triangle, the opposite side is 11 cm and the adjacent side is 7 cm. Find the angle \theta.

  1. \tan\theta=11/7.
  2. \theta=\tan^{-1}(11/7).
  3. \theta\approx57.529^\circ.

Answer: 57.5^\circ to 1 decimal place.

Angles Of Elevation And Depression

An angle of elevation is measured upward from a horizontal line; an angle of depression is measured downward. Horizontal lines are parallel, so alternate angles may allow the same angle to be placed inside the right triangle.

Worked Example: Height From An Angle

Question: A point is 30 m from the base of a vertical tower. The angle of elevation to the top is 41^\circ. Find the height.

  1. Opposite =h, adjacent =30.
  2. \tan41^\circ=h/30.
  3. h=30\tan41^\circ\approx26.08.

Answer: 26.1 m to 3 significant figures.

Combining Pythagoras And Trigonometry

A problem may require Pythagoras to find a missing side before using a trigonometric ratio, or vice versa. Draw the relevant right triangle and keep intermediate values unrounded.

Scope Note

This D2 chapter covers right-angled triangles and basic two-dimensional applications. The sine rule, cosine rule, triangle area formula, bearings and more advanced applications are developed in later 8th Edition chapters.

Examination Guidance
  • Write SOH CAH TOA or the three formulas before selecting a ratio.
  • Check that the calculator is in degree mode.
  • Use inverse trig only when the unknown is an angle.
  • Keep full calculator precision until the final answer.
Common Mistakes
  • Labelling the hypotenuse as adjacent.
  • Choosing opposite and adjacent without reference to the stated angle.
  • Using \sin^{-1}x to mean 1/\sin x; in this context it means inverse sine.
  • Rounding an intermediate side too early.
Knowledge Check And Practice

1. A right triangle has hypotenuse 20 cm and angle 30^\circ. Find the opposite side.

Answer: 20\sin30^\circ=10 cm.

2. Adjacent side is 12 m and hypotenuse 15 m. Find the angle.

Answer: \cos^{-1}(12/15)\approx36.9^\circ.

3. Opposite side is 8 cm and adjacent side is 6 cm. Find the angle.

Answer: \tan^{-1}(8/6)\approx53.1^\circ.

4. Which ratio uses adjacent and hypotenuse?

Answer: Cosine.

5. A ladder 5 m long reaches 4.3 m up a wall. Find its angle with the ground.

Answer: \sin^{-1}(4.3/5)\approx59.3^\circ.
Extended Worked Practice
Angle Of Elevation With Eye Height

Question: A student 1.6 m tall stands 25 m from a tower. The angle of elevation from the student’s eye to the top is 38 degrees. Find the tower height.

  1. Height above eye level =25\tan38^\circ.
  2. 25\tan38^\circ\approx19.53 m.
  3. Add eye height: 19.53+1.6=21.13 m.

Answer: 21.1 m to 3 significant figures.

Finding An Angle From Two Sides

Question: In a right triangle, the side opposite \theta is 12 cm and the adjacent side is 19 cm. Find \theta.

\tan\theta=12/19, so \theta=\tan^{-1}(12/19)\approx32.3^\circ.

Guy Wire And Pole

Question: A 14 m guy wire is attached to the top of a vertical pole and makes an angle of 62 degrees with the ground. Find the pole height and horizontal distance from the pole to the anchor.

Height =14\sin62^\circ\approx12.4 m. Horizontal distance =14\cos62^\circ\approx6.57 m.

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