Grade by Grade
C. Trigonometric Functions · Trigonometry of Acute Triangles

Finding Angles and Solving Right Triangles

C1.1About 70 minutes
Learning goals
  • I can use sin⁻¹, cos⁻¹ and tan⁻¹ to find an acute angle from two known sides.
  • I can "solve" a right triangle: find every unknown side and angle.
  • I can use the angle sum (the two acute angles add to 90°) and the Pythagorean theorem to finish and check a solution.
  • I can find slope angles of ramps, roofs and hills.

Words to know

inverse trigonometric ratio
sin⁻¹, cos⁻¹ or tan⁻¹. It takes a ratio and gives back the angle. For example, sin⁻¹(0.5) = 30°.
solve a triangle
Find all the unknown side lengths and angle measures.
complementary angles
Two angles that add to 90°. The two acute angles of a right triangle are complementary.
pitch (of a roof)
Rise over run, often written like 6/12: the roof rises 6 units for every 12 units across.

From sides back to angles

In the last lesson you knew an angle and used it to find a side. Very often it is the other way around. A carpenter knows a roof rises 6 cm for every 12 cm of run; a builder knows a ramp rises 1 cm for every 12 cm across. What angle is that?

The calculator's inverse keys answer this. If sin θ = 0.5, then θ = sin⁻¹(0.5) = 30°. Read sin⁻¹ as "the angle whose sine is". (The −1 is not an exponent here: sin⁻¹ x does not mean 1 ÷ sin x.)

θACB12 (adjacent)5 (opp)13 (hypotenuse)θ = sin⁻¹(5/13) ≈ 22.6°θ = cos⁻¹(12/13) ≈ 22.6°θ = tan⁻¹(5/12) ≈ 22.6°Any pair of sides givesthe same angle.
In a 5-12-13 right triangle, all three inverse ratios give the same angle θ ≈ 22.6°. Use whichever pair of sides you actually know.

Inverse ratios

  • If you know opposite and hypotenuse: θ = sin⁻¹(opp/hyp)
  • If you know adjacent and hypotenuse: θ = cos⁻¹(adj/hyp)
  • If you know opposite and adjacent: θ = tan⁻¹(opp/adj)

On most calculators, press 2nd (or SHIFT) then sin to get sin⁻¹. Round angles to the nearest degree unless the question asks for more (slopes and navigation often use one decimal place).

Worked example

Example 1: How steep is the ramp?

A ramp rises 1 cm for every 12 cm of horizontal run. What angle does it make with the ground?

  1. Rise is opposite the angle; run is adjacent. Use tangent.
  2. tan θ = 1/12 = 0.0833...
  3. θ = tan⁻¹(1/12) = 4.764°...

The ramp makes an angle of about 4.8° with the ground.

Solving a right triangle

To solve a triangle means to find all three sides and all three angles. In a right triangle you already know one angle (90°), so you need just two more facts, and at least one of them must be a side. Two shortcuts help:

  • The two acute angles are complementary: they add to 90°.
  • The Pythagorean theorem gives the third side when two sides are known.

When you can, find each new value from the given information rather than from a value you just rounded.

What do I know?Two sides1. Inverse ratio → one angle2. Other angle = 90° − first3. Pythagoras → third sideOne side + one acute angle1. Other angle = 90° − given2. A ratio → second side3. Another ratio → third side
Two starting points for solving a right triangle. Either way you finish with all three sides and all three angles.
Worked example

Example 2: Solve from two sides

In △ABC, ∠C = 90°, a = 7.5 cm and b = 11.2 cm. Solve the triangle.

  1. Third side: c = √(7.5² + 11.2²) = √181.69 ≈ 13.5 cm.
  2. From ∠A, a is opposite and b is adjacent: tan A = 7.5/11.2.
  3. ∠A = tan⁻¹(7.5/11.2) ≈ 33.8°, so about 34°.
  4. ∠B = 90° − 33.8° ≈ 56.2°, about 56°.
  5. Check: the longer leg b (11.2) is across from the larger angle B. ✓

c ≈ 13.5 cm, ∠A ≈ 34°, ∠B ≈ 56°

Worked example

Example 3: Solve from a side and an angle

In △PQR, ∠Q = 90°, ∠P = 41° and the hypotenuse PR = 25 m. Solve the triangle.

  1. ∠R = 90° − 41° = 49°.
  2. QR is opposite ∠P: QR = 25 sin 41° ≈ 16.4 m.
  3. PQ is adjacent to ∠P: PQ = 25 cos 41° ≈ 18.9 m.
  4. Check: 16.4² + 18.9² ≈ 626 ≈ 25² = 625. ✓

∠R = 49°, QR ≈ 16.4 m, PQ ≈ 18.9 m

Worked example

Example 4: Roof pitch

A garage roof has a 6/12 pitch: it rises 6 units for every 12 units of horizontal run (for example, 6 cm for every 12 cm). What angle does the roof make with the horizontal? Steeper roofs shed snow more easily; is this roof steeper than 30°?

  1. tan θ = rise/run = 6/12 = 0.5.
  2. θ = tan⁻¹(0.5) ≈ 26.6°.

The roof angle is about 26.6°, which is less than 30°.

Just for fun

Why are inverse trig keys so calm?

Because whatever ratio you throw at them, they always know what angle to take.

Hands-on activity

How steep are your stairs?

You need: a ruler or tape measure, a staircase (or a ramp, or a sloped driveway), paper and a calculator.

  1. Measure the rise (height) of one step and its run (depth of the tread, where your foot goes).
  2. Calculate the angle of the staircase: θ = tan⁻¹(rise ÷ run).
  3. Measure three different steps and average the angles.
  4. If you have a ramp or driveway nearby, measure a vertical drop over a measured horizontal distance (hold a level or a straight board horizontally) and find its angle too.

What to look for: home stairs often come out somewhere between 30° and 45°. A ramp should be a much smaller angle (often under 5°). Write one sentence comparing them.

Your turn

1

Find θ if sin θ = 0.6157.

2

Find θ if cos θ = 3/8.

3

A right triangle has opposite side 9 cm and adjacent side 14 cm (relative to θ). Find θ.

4

A 5.0 m ladder reaches 4.6 m up a wall. What angle does it make with the ground?

5

Solve △ABC: ∠C = 90°, a = 16 m, c = 30 m.

6

Solve △DEF: ∠E = 90°, ∠D = 58°, DE = 6.4 cm.

7

A road rises 85 m over a horizontal distance of 1.2 km. What angle does it make with the horizontal? (Watch the units.)

More practice: warm-up

1

What does sin⁻¹(0.5) mean in words?

2

Find tan⁻¹(1).

3

The acute angles of a right triangle are x and 37°. Find x.

4

Find θ if tan θ = 2.5.

5

Find the hypotenuse of a right triangle with legs 9 and 12.

More practice: on your own

1

A snowboard ramp is 4.0 m long along its slope and rises 1.1 m. Find its angle with the ground.

2

Solve △XYZ: ∠Z = 90°, x = 21 cm, y = 20 cm.

3

Mia found ∠A = 34° and then computed the opposite side using 34° rounded from 33.6°. Her final answer was off by 0.1. Explain why, and what to do instead.

4

The diagonal of a rectangular field is 130 m and one side is 50 m. Find the angle between the diagonal and the 50 m side, and the other side length.

5

A roof has a pitch of 8/12. A second roof has an angle of 35°. Which is steeper?

Challenge

1

In right △ABC (∠C = 90°), sin A = 0.8. Without finding the angle, find cos A and tan A. (Hint: pick a hypotenuse of 10.)

2

Explain why sin⁻¹(1.2) gives an error on your calculator.

3

A rectangle is 18 cm by 7 cm. Find the acute angle between its two diagonals.

Check your understanding

Quiz

1.

What does cos⁻¹(0.5) equal?

2.

A right triangle has opposite 7 and adjacent 24 for angle θ. Which gives θ?

3.

One acute angle of a right triangle is 28°. What is the other?

4.

A ramp rises 1 cm for every 12 cm of run. Its angle is about:

5.

Which statement about sin⁻¹ x is true?

6.

In △ABC, ∠C = 90°, a = 7.5 and b = 11.2. What is ∠A to the nearest degree?

For parents and teachers

Big idea: This lesson completes the right-triangle part of expectation C1.1: finding angles with inverse ratios and solving whole right triangles. It also uses ramps, roofs and stairs as real-world contexts.

How to help: When your teen finds an angle, ask: "Does that angle look right in your sketch?" A 70° answer for a gentle ramp should set off alarm bells.

Common difficulties:

  • Pressing sin instead of 2nd/SHIFT + sin, or thinking sin⁻¹ x means 1 ÷ sin x.
  • Using the hypotenuse with tan⁻¹ (tangent uses the two legs).
  • Mixing units (metres and kilometres) in one ratio.
  • Rounding an angle and then using the rounded value in the next step.