10.05 Solving for angles in right triangles
Inverse trigonometric ratios
Just like the how our basic operations like addition and subtraction have inverse operations that "undo" them, our three trigonometric ratios, sine, cosine and tangent, also have inverse operations.
Given a trigonometric ratio for an angle, we want to find the measure of the related angle.
Exploration
Suppose we are told that $\sin\theta=\frac{12}{13}$sinθ=1213 and asked to find $m\angle\theta$m∠θ. How would we do this if all we had was a ruler and a protractor?
Well, we could start by recognizing that this must be a triangle with a hypotenuse of length $13$13 and another side of length $12$12. We could use the Pythagorean theorem to find the third side (or recognize the Pythagorean triple). We could then perfectly draw out the triangle below.
We could finally use our protractor to measure $\angle\theta$∠θ. We would find that it is about $67^\circ$67°.
Now that took a long time and is not the most accurate. Is there a better way?
Calculating the inverse using a calculator
As with every topic in mathematics, there is a conceptual side (what you need to know and understand - seen above) and a practical side (what you need to do and answer). To calculate values involving trigonometric expressions, it will often be easiest to use a scientific calculator.
Worked example
Question 1
If $\sin\theta=\frac{12}{13}$sinθ=1213, find $\theta$θ to the nearest degree.
Think: This question is asking us what the angle ($\theta$θ) is, if the ratio of the opposite side and hypotenuse is $\frac{12}{13}$1213. To answer this question, we can use the inverse sine button on a calculator. It will probably look like $\sin^{-1}$sin−1, and may involving pressing 'shift' or '2nd F'.
Do:
| $\sin\theta$sinθ | $=$= | $\frac{12}{13}$1213 | |
| $m\angle\theta$m∠θ | $=$= | $\sin^{-1}\left(\frac{12}{13}\right)$sin−1(1213) | (Take the inverse sine) |
| $\theta$θ | $=$= | $67.380135$67.380135$\ldots$… | (Evaluate with a calculator) |
| $\theta$θ | $=$= | $67^\circ$67° | (Round to the nearest degree) |
Reflect: This is the same answer we got when drawing out the triangle.
Practice questions
QUESTION 2
If $\cos\theta=0.146$cosθ=0.146, find $\theta$θ, writing your answer to the nearest degree.
QUESTION 3
If $\tan\theta=1.711$tanθ=1.711, find $\theta$θ, writing your answer to the nearest degree.
Solving for unknown angles
We can also use the trigonometric ratios to find the size of unknown angles. To do this we need any $2$2 of the side lengths.
- Label the sides as O, A or H with respect to the position of the angle you want to find
- Identify the appropriate trigonometric ratio that applies [either sine (sin), cosine (cos) or tangent (tan)]
- Using algebra, solve the equation for the angle, (write the rule, fill in what you know, then solve using inverse operations)
- Reflect and check (do a quick check on your calculator to confirm your answer is correct)
Worked examples
Question 4
Find the measure of the angle $\theta$θ in the diagram below to two decimal places.
Think: We first need to label the sides as O, A or H with respect to the position of the angle to identify the appropriate ratio. We have the opposite and hypotenuse, so we need to use the sine ratio.
Do:
| $\sin\theta$sinθ | $=$= | $\frac{O}{H}$OH |
State the formula for the correct ratio |
| $\sin\theta$sinθ | $=$= | $\frac{5}{8}$58 |
Fill in the given information |
| $\theta$θ | $=$= | $\sin^{-1}\left(\frac{5}{8}\right)$sin−1(58) |
Use the inverse ratio to solve for $\theta$θ |
| $\theta$θ | $=$= | $38.68^\circ$38.68° |
Use a calculator to evaluate |
Question 5
Find the value of the angle indicated to two decimal places.
Think: We have the opposite and adjacent sides here, so the ratio we will use is tangent (tan).
Do:
| $\tan\theta$tanθ | $=$= | $\frac{O}{A}$OA |
State the formula for the correct ratio |
| $\tan\theta$tanθ | $=$= | $\frac{14.77}{12.24}$14.7712.24 |
Fill in the given information |
| $\theta$θ | $=$= | $\tan^{-1}\left(\frac{14.77}{12.24}\right)$tan−1(14.7712.24) |
Use the inverse ratio to solve for $\theta$θ |
| $\theta$θ | $=$= | $50.35^\circ$50.35° |
Use a calculator to evaluate |
Practice questions
Question 6
Use the tangent ratio to find the size of the angle marked $x$x, correct to the nearest degree.
Question 7
The person in the picture sights a pigeon above him. If the angle the person is looking at is $\theta$θ, find $\theta$θ in degrees.
Round your answer to two decimal places.
Exact value triangles
Just as we can solve for the sides of triangles involving exact values from our special triangles, we can solve for angles and get $30^\circ$30°, $45^\circ$45° or $60^\circ$60° as exact answers instead of the irrational values in the previous examples.
Remember our special triangles:
The table below is another way to display the information in the exact value triangles. You can choose which method you prefer to help you remember these exact ratios.
| sin | cos | tan | |
|---|---|---|---|
| $30^\circ$30° | $\frac{1}{2}$12 | $\frac{\sqrt{3}}{2}$√32 | $\frac{1}{\sqrt{3}}$1√3 |
| $45^\circ$45° | $\frac{1}{\sqrt{2}}$1√2 | $\frac{1}{\sqrt{2}}$1√2 | $1$1 |
| $60^\circ$60° | $\frac{\sqrt{3}}{2}$√32 | $\frac{1}{2}$12 | $\sqrt{3}$√3 |
Practice questions
QUESTION 8
Given that $\sin\theta=\frac{1}{2}$sinθ=12, we want to find the value of $\cos\theta$cosθ.
First, find the value of $\theta$θ.
Hence, find the exact value of $\cos30^\circ$cos30°.
Question 9
You are given that $\tan\theta=\frac{1}{\sqrt{3}}$tanθ=1√3.
First, find the value of $\theta$θ.
Hence, find exact the value of $\sin\theta$sinθ.
Question 10
Given that $\cos\theta=\frac{1}{\sqrt{2}}$cosθ=1√2, we want to find the value of $\tan\theta$tanθ.
First find the value of $\theta$θ.
Hence, find the exact value of $\tan45^\circ$tan45°.