Thirty-Six

Unit 7 · TrigonometryG6 · Trigonometry · Aerodynamics & Navigation

Right-Triangle Trigonometry

Choose sine, cosine, or tangent and solve right triangles without ratio mix-ups.

  • 28 min
  • Unlocks: Star Tracker
  • Desmos way

Ready when you are

7 steps · about 28 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

sine
Opposite leg divided by hypotenuse.
cosine
Adjacent leg divided by hypotenuse.
tangent
Opposite leg divided by adjacent leg.

Start by naming the quantity each problem asks for. That habit blocks the most common ACT trap.

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Try it first

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Take a calm first shot. Label what you know, write one relationship, and make the best choice you can.

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The big idea

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Build the picture before the arithmetic

Circle the angle, mark O, A, and H, then choose the ratio containing the known side and the side you need. Inverse trig finds an angle; ordinary trig finds a side.

Key idea

The decision that matters

Label the sides from the marked angle first. The hypotenuse is always across from the right angle; opposite and adjacent depend on where you stand.

SOH-CAH-TOA
sin⁡θ=oppositehypotenuse,cos⁡θ=adjacenthypotenuse,tan⁡θ=oppositeadjacent\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\qquad \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\qquad \tan\theta=\frac{\text{opposite}}{\text{adjacent}}

Write this relationship before inserting numbers. It keeps labels and units attached to the work.

A compact example

A right triangle has legs 9 and 12, so its hypotenuse is 15. From the angle opposite the 9-unit leg, sin⁡θ=915=35\sin\theta=\frac{9}{15}=\frac35.

Explore in Desmos

Check an angle-of-elevation calculation

Keep the calculator in degree mode. Read the height for a 48-foot ground distance and a 32° angle, then change 32° to 25° and 40°. Record how the height changes while the ground distance stays fixed.

The calculator is live: change anything and see what happens.

Check an angle-of-elevation calculation

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

Loading Desmos…
CueBest move
Need a sideChoose the ratio containing that side
Need an angleUse inverse trig after forming a side ratio

Trap

Choosing a ratio before labeling the sides

This error produces a believable answer. Pause before computing and say what each number means.

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Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1: Find a trig ratio

Problem

A right triangle has legs 8 and 15 and hypotenuse 17. For the angle θ\theta opposite the 8-unit leg, find cos⁡(θ)\cos(\theta).
From θ\theta, the adjacent leg is 15 and the hypotenuse is 17.

Cosine uses adjacent over hypotenuse: cos⁡(θ)=1517\cos(\theta)=\frac{15}{17}.

Which ratio gives cosine from the marked angle?

The ratio is below 1, as every acute-angle cosine must be.

Why is 15 over 17 the cosine ratio?

Worked example

Example 2: Height from an angle of elevation

Problem

A surveyor stands 48 feet from a flagpole. The angle of elevation to the top is 32∘32^\circ. Find the pole height to the nearest foot.
The 48-foot distance is adjacent to 32∘32^\circ; height hh is opposite.

Use tangent: tan⁡(32∘)=h/48\tan(32^\circ)=h/48.

Which equation isolates the pole height?

Multiply: h=48tan⁡(32∘)≈29.99h=48\tan(32^\circ)\approx29.99, so the pole is about 30 feet tall.

In Desmos degree mode, type 48tan(32). The readout is about 29.99.

Why does tangent model the pole height?

Faded example · you finish it

Example 3: You finish it

Problem

A 55-1212-1313 right triangle has θ\theta opposite the 5-unit leg. Complete sin⁡(θ)=__/__\sin(\theta)=\_\_/\_\_.
Label opposite, adjacent, and hypotenuse from the marked angle before choosing a ratio.

Which completion is correct?

Which reason validates the result?

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Practice

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Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.