So far we have looked at the relationship between angles and side lengths in right-angled triangles using the ratios $\sin$sin, $\cos$cos and $\tan$tan. We know how to calculate the ratios, and how to use them to find the length of unknown sides of right-angled triangles.
For any angle theta, $\sin\theta$sinθ, $\cos\theta$cosθ and $\tan\theta$tanθ give us the ratio that corresponds to that angle.
Trigonometric ratios can also be used to find the size of unknown angles. To do this we need any $2$2 of the side lengths in a right-angled triangle.
We can then do the following:
- Label the sides as opposite, adjacent or hypotenuse with respect to the angle we want to find.
- Identify the trigonometric ratio of this angle which involves the known lengths of $2$2 sides.
- Write and solve the equation for this ratio of the unknown angle.
For example, in this triangle we can see that the $\sin$sin ratio of theta is $0.5$0.5. So we can form the equation $\sin\theta=0.5$sinθ=0.5 and use the inverse $\sin$sin function to solve for the angle $\theta$θ.
For the last step, we need to be able to 'undo' the trigonometric ratio of an angle. We do this by applying an inverse trigonometric ratio to both sides of the equation. These are $\sin^{-1}$sin−1, $\cos^{-1}$cos−1, and $\tan^{-1}$tan−1 and they are defined so that:
If $\cos\theta=y$cosθ=y, then $\cos^{-1}(y)=\theta$cos−1(y)=θ.
If $\tan\theta=z$tanθ=z, then $\tan^{-1}(z)=\theta$tan−1(z)=θ.
In the case of the above example, if $\sin\theta=0.5$sinθ=0.5, then $\theta=\sin^{-1}(0.5)$θ=sin−1(0.5).
Finding the inverse with a calculator
We can use these inverse trigonometric functions by pressing the appropriate button on a scientific calculator. The buttons usually appear next to or above the $\sin$sin, $\cos$cos and $\tan$tan buttons.
If you enter $\sin^{-1}(0.5)$sin−1(0.5) in your calculator, you get the result $30^\circ$30°. This tells us that an angle of $30^\circ$30° has a sine ratio of $0.5$0.5.
The ratio $\tan\theta$tanθ can be any real number, so $\tan^{-1}$tan−1 can take any value and return the corresponding angle.
Example 1
Find the angle $\theta$θ in the triangle below to the nearest degree.
1. Label the sides as opposite, adjacent or hypotenuse with respect to $\theta$θ.
2. Identify the ratio that relates the opposite side and the hypotenuse. For this angle, it will be $\sin$sin.
3. Write and solve the equation for $\sin\theta$sinθ.
| $\sin\theta$sinθ | $=$= | $\frac{\text{Opposite }}{\text{Hypotenuse }}$Opposite Hypotenuse |
(write the rule) |
| $\sin\theta$sinθ | $=$= | $\frac{5}{8}$58 |
(fill in the known side lengths) |
| $\theta$θ | $=$= | $\sin^{-1}\left(\frac{5}{8}\right)$sin−1(58) |
('undo' the $\sin$sin function) |
| $\theta$θ | $=$= | $39^\circ$39° |
(use a calculator to evaluate $\theta$θ to the nearest degree) |
Example 2
Find the angle $\theta$θ in the triangle below to the nearest degree.
In this triangle, we know the lengths of the the opposite and adjacent sides. So the ratio we will use is $\tan$tan.
| $\tan\theta$tanθ | $=$= | $\frac{\text{Opposite }}{\text{Adjacent }}$Opposite Adjacent |
(write the rule) |
| $\tan\theta$tanθ | $=$= | $\frac{14.77}{12.24}$14.7712.24 |
(fill in the known side lengths) |
| $\theta$θ | $=$= | $\tan^{-1}\left(\frac{14.77}{12.24}\right)$tan−1(14.7712.24) |
('undo' the $\tan$tan function) |
| $\theta$θ | $=$= | $50^\circ$50° |
(use a calculator to evaluate $\theta$θ to the nearest degree) |
Example 3
Find the value of $\theta$θ to the nearest degree.
Example 4
Consider the given figure.
Find $\theta$θ, correct to the nearest degree.
Find $\alpha$α, correct to the nearest degree.
Find $\beta$β, correct to the nearest degree.
Example 5
Lachlan spots a paraglider above him. If the angle the Lachlan is looking at is $\alpha$α, find $\alpha$α to the nearest degree.
