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Trigonometry · Trigonometry: Similar Triangles, Right Triangles and Acute Triangles

Using Inverse Trigonometric Ratios to Find Angles

2.2 2.3About 65 minutes
Learning goals
  • I can use sin⁻¹, cos⁻¹ and tan⁻¹ on a calculator to find an angle from a ratio.
  • I can find an acute angle in a right triangle when I know two sides.
  • I can check that an angle answer is reasonable.

Words to know

inverse sine (sin⁻¹)
Gives the angle whose sine is a given number. sin⁻¹(0.5) = 30°.
inverse cosine (cos⁻¹)
Gives the angle whose cosine is a given number.
inverse tangent (tan⁻¹)
Gives the angle whose tangent is a given number.
slope / grade
Rise ÷ run. A grade of 8% means a rise of 8 m for every 100 m of horizontal distance.

Going backwards

So far we started with an angle and found a side. But often you know the sides and want the angle: How steep is this ramp? What angle does this roof make? At what angle is the ladder leaning? For that we run the trig ratios backwards, using the inverse ratios sin⁻¹, cos⁻¹ and tan⁻¹.

angle: 30°sinratio: 0.5ratio: 0.5sin⁻¹angle: 30°sin turns an angle into a ratio; sin⁻¹ turns a ratio back into an angle.
A trig ratio and its inverse undo each other, just like squaring and taking a square root.

On the calculator

The inverse keys are usually the 2nd (or SHIFT) function of the sin, cos and tan keys. To find the angle whose sine is 0.5, press 2nd, then sin, then 0.5: sin⁻¹(0.5) = 30°.

Warning: sin⁻¹ does not mean 1 ÷ sin. The −1 is a symbol for "inverse", not an exponent. Also, since sin and cos of an acute angle are always less than 1, sin⁻¹(1.2) gives an error: no angle has a sine of 1.2. If you see that error, you probably put the hypotenuse on top.

Worked example

Example 1: Using sin⁻¹

In a right triangle, the side opposite θ is 5 cm and the hypotenuse is 13 cm. Find θ to the nearest degree.

  1. Opposite and hypotenuse → sin. sin θ = 5/13.
  2. θ = sin⁻¹(5/13) = 22.6...°.
  3. Round: θ ≈ 23°.

θ ≈ 23°

Worked example

Example 2: Using cos⁻¹

The side adjacent to θ is 7 m and the hypotenuse is 10 m. Find θ.

  1. cos θ = 7/10 = 0.7.
  2. θ = cos⁻¹(0.7) = 45.57...° ≈ 46°.

θ ≈ 46°

Worked example

Example 3: Using tan⁻¹

The side opposite θ is 9 cm and the adjacent side is 4 cm. Find θ and the other acute angle.

  1. tan θ = 9/4 = 2.25.
  2. θ = tan⁻¹(2.25) = 66.03...° ≈ 66°.
  3. The other acute angle is 90° − 66.03...° ≈ 24°. (The two acute angles of a right triangle add to 90°.)
  4. Check: the opposite side is longer than the adjacent side, so θ should be more than 45°. ✓

θ ≈ 66°, and the other angle is about 24°.

9 cm4 cmθopposite = 9, adjacent = 4tan θ = 9/4 = 2.25θ = tan⁻¹(2.25) ≈ 66°Check: θ > 45° becausethe opposite side is longer.
Example 3, drawn to scale. A long opposite side and a short adjacent side give a steep angle.
Worked example

Example 4: How steep is the ramp?

A ramp rises 1 m over a horizontal distance of 12 m. What angle does it make with the ground, to the nearest tenth of a degree?

  1. Rise is opposite θ; horizontal run is adjacent. tan θ = 1/12.
  2. θ = tan⁻¹(1/12) = 4.76...° ≈ 4.8°.

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

θhorizontal run 12 m1 mramp surface
A rise of 1 m over a 12 m run, drawn to scale. A gentle ramp like this makes an angle of less than 5°.

Checking angle answers

  • Both acute angles in a right triangle are between 0° and 90° and add to 90°.
  • The bigger angle is opposite the longer leg.
  • If the two legs are equal, both angles are 45°. So if the opposite side is shorter than the adjacent side, θ < 45°; if it is longer, θ > 45°.
Just for fun

sin⁻¹ has another name: arcsin (and cos⁻¹ is arccos, tan⁻¹ is arctan). It sounds like the name of a cartoon villain, but all it ever does is find angles.

Hands-on activity

Measure the steepness of your stairs

You need: a ruler or measuring tape, a calculator, and a staircase (indoors or outdoors). Be careful on the stairs.

  1. Measure the rise of one step (its height) and the run (the depth of the flat part you step on). Measure several steps and average.
  2. Calculate the angle of the stairs: θ = tan⁻¹(rise ÷ run).
  3. Find a ramp, a driveway or a slide and measure a rise and a run there too (for a slide or a driveway, measure the run along level ground with a helper).
  4. Rank them from gentlest to steepest.

What to look for: home stairs often come out somewhere between about 30° and 40°, which is much steeper than a ramp. For example, a rise of 18 cm and a run of 26 cm gives tan⁻¹(18 ÷ 26) ≈ 35°.

Your turn

1

Find θ if sin θ = 0.4 (nearest degree).

2

Find θ if cos θ = 0.25.

3

Find θ if tan θ = 3.5.

4

Opposite = 8 and hypotenuse = 11. Find θ.

5

Adjacent = 6 and hypotenuse = 15. Find θ.

6

Opposite = 10 and adjacent = 7. Find θ.

7

A 6 m ladder leans against a wall with its foot 1.5 m from the wall. What angle does it make with the ground?

More practice: warm-up

1

sin⁻¹(0.5) = ?

2

cos⁻¹(0.5) = ?

3

tan⁻¹(1) = ?

4

Is sin⁻¹(0.5) the same as 1 ÷ sin(0.5)? Explain.

5

tan θ = 0.75. Find θ to the nearest degree.

More practice: on your own

1

In △ABC, ∠C = 90°, AC = 12 cm and BC = 5 cm. Find ∠A and ∠B.

2

A roof rafter rises 2.4 m over a horizontal run of 4.5 m. What angle does the roof make with the horizontal?

3

A 20 m support wire runs from the top of a 15 m vertical pole to the ground. What angle does the wire make with the ground?

4

A road sign warns of an 8% grade (8 m rise for every 100 m horizontal). What angle does the road make with the horizontal?

5

Mia typed sin⁻¹(13/5) and got an error for a triangle with opposite 5 and hypotenuse 13. Explain and fix it.

Challenge

1

A common ladder-safety rule of thumb is to place the foot of the ladder 1 m out from the wall for every 4 m of height. What angle does that make with the ground?

2

An isosceles triangle has a base of 10 cm and two equal sides of 13 cm. Find all three angles to the nearest tenth of a degree.

3

A rectangle is 8 cm by 5 cm. Find the acute angle between its two diagonals.

Check your understanding

Quiz

1.

sin θ = 0.5. What is θ?

2.

In a right triangle the two legs are both 7 cm. What is each acute angle?

3.

You know the adjacent side and the hypotenuse. How do you find the angle?

4.

Why does sin⁻¹(1.2) give an error?

5.

Opposite = 3 and hypotenuse = 5. What is θ to the nearest degree?

6.

A ramp rises 0.5 m over a horizontal run of 6 m. About what angle does it make with the ground?

For parents and teachers

Big idea: Expectations 2.2 and 2.3: determining angle measures in right triangles with the inverse ratios, and using them in everyday situations like ramps, roofs and ladders.

How to help: Make sure your teen can find the inverse keys (usually 2nd or SHIFT). The stair-measuring activity takes ten minutes and connects the math to real objects. Ask "Does the angle make sense? Is it steep or gentle?"

Common difficulties:

  • Reading sin⁻¹ as 1 ÷ sin.
  • Putting the fraction upside down (getting an error for sin⁻¹ or cos⁻¹).
  • Rounding the ratio before using the inverse key (use the full fraction, such as sin⁻¹(5 ÷ 13)).
  • Forgetting the quick check that the two acute angles add to 90°.