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Trigonometry

Geometry · Trigonometry

Syllabus tag: Kenya CBC | Grade 9 Mathematics | Strand 4.0 Geometry | Sub-Strand 4.4 Trigonometry (7 Lessons)

Lesson objectives

By the end of this sub-strand, you should be able to:

  • Identify the hypotenuse, opposite and adjacent sides relative to a given angle.
  • State and apply the sine, cosine and tangent ratios in right-angled triangles.
  • Use trigonometric tables or a calculator to find sides and angles.
  • Apply trigonometry to problems involving angles of elevation and depression.

Trigonometry

You cannot climb a tall tree to measure it. You cannot swim across a river with a tape measure. Trigonometry lets you find these lengths from the ground.

It works on one shape only: the right-angled triangle. If you can find a right angle in a problem, you can usually solve it.

a) Naming the sides

Every right-angled triangle has three sides. Two of the names depend on which angle you are working from.

The three sides, named from the angle The three sides, named from the angle θ adjacent opposite hypotenuse

The hypotenuse is always the longest side. It sits opposite the right angle, and its name never changes.

The opposite side is the one directly across from your angle. The adjacent side is the one touching your angle, not counting the hypotenuse.

Pick a different angle and the opposite and adjacent swap places. The hypotenuse stays put.

b) The three ratios

A ratio compares two sides by dividing one by the other. There are three you need.

The sine ratio The sine ratio sin θ = opposite hypotenuse
The cosine ratio The cosine ratio cos θ = adjacent hypotenuse
The tangent ratio The tangent ratio tan θ = opposite adjacent

Many learners remember these as SOH CAH TOA. Say it out loud a few times. Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.

c) Choosing which ratio to use

This is the step most learners find hard. The method is simple.

Look at the two sides in your question. One will be known, and one will be what you want. Then read off the ratio that uses exactly those two sides.

Sides involvedUse
Opposite and hypotenusesine
Adjacent and hypotenusecosine
Opposite and adjacenttangent

You never need to guess. The pair of sides tells you the answer.

d) Finding a side you do not know

:::example Worked example 1 -- how tall is the tree?

You stand 20 metres from a tall tree. Looking up at the top, your line of sight makes an angle of 35 degrees with the ground. How tall is the tree? Take sin 35 as 0.5736 and tan 35 as 0.7002.

Finding a height you cannot reach Finding a height you cannot reach 35° 20 m h line of sight
Finding the height Finding the height tan θ = opposite / adjacent write down the rule tan 35° = h / 20 put in what you know 20 × tan 35° = h multiply both sides by 20 h = 20 × tan 35° turn it round, unknown first h = 20 × 0.7002 tan 35° is 0.7002 h = 14.004 do the multiplication h = 14.0 m round to one decimal place

The tree is about 14 m tall. Shorter than the 20 m you stood back, as it should be. :::

e) Finding an angle you do not know

Sometimes you know two sides and want the angle. You form the ratio in the usual way, then work backwards using the inverse key on your calculator.

:::example Worked example 2 -- the safe ladder

A ladder 6 metres long leans against a wall. Its foot is 2.4 metres from the base of the wall. What angle does it make with the ground?

The ladder problem The ladder problem θ 2.4 m along ground ladder, 6 m

The ladder is the hypotenuse at 6 m. The 2.4 m along the ground is adjacent.

Finding the angle Finding the angle cos θ = adjacent / hypotenuse write down the rule cos θ = 2.4 / 6 put in what you know cos θ = 0.4 do the division θ = cos⁻¹ 0.4 inverse cosine undoes cosine θ = 66.42° read it off the calculator θ = 66° round to the nearest degree

Ladder safety asks for about 75 degrees. At 66 degrees this one is too shallow, and its foot could slide out. :::

One check is always worth doing. Sine and cosine can never be larger than 1, because the hypotenuse is the longest side. If you get 1.4, you have paired the sides the wrong way round.

f) Where this is used

Surveyors setting out plots in Kajiado use these ratios every day. So do builders working out roof slopes, and engineers checking the gradient of a road climbing an escarpment.

Words to know

  • Hypotenuse -- the longest side of a right-angled triangle, opposite the right angle.
  • Opposite side -- the side directly across from the angle being considered.
  • Adjacent side -- the side between the chosen angle and the right angle.
  • Angle of elevation -- the angle measured upward from the horizontal to a line of sight.
  • Angle of depression -- the angle measured downward from the horizontal to a line of sight.

:::checkpoint Check yourself

  1. In a right-angled triangle, which side is never called opposite or adjacent?
  2. You know the opposite side and the hypotenuse. Which ratio do you use?
  3. A learner calculates sin θ = 1.6. What has gone wrong?
  4. A pole is 12 m tall. From a point on the ground the angle of elevation to its top is 40 degrees. Which ratio finds the distance from the pole? :::

Bridge to practice

The exercises begin with naming sides and selecting the right ratio, move through finding unknown sides and angles, and finish with elevation and depression problems including one requiring eye height to be added. For every question, sketch the triangle and mark the angle before choosing an option, and check that any sine or cosine you compute is less than 1.

Check yourselfPractise Trigonometry — 10 questions →Next in MathematicsMoney →