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Summary
Calculating the missing angle of a right-angled triangle is very similar to calculating the missing length. However, it is important to first learn about inverse trigonometric functions.
Inverse trigonometric functions invert the trigonometric function, giving the value of the angle $x$.
Function | What it's called | examples |
$\sin^{-1}()$ | Inverse sine | If $\sin(x) = 0.5$, then $x = \sin^{-1}(0.5)$ |
$\cos^{-1}()$ | Inverse cosine | If $\cos(x) = 0.8$, then $x = \cos^{-1}(0.8)$ |
$\tan^{-1}()$ | Inverse tangent | If $\tan(x) = 0.42$, then $x = \tan^{-1}(0.42)$ |
Note: The inverse function $\sin^{-1}()$ is NOT the same as $\frac{1}{\sin()}$! It is just notation to tell us that it is the inverse of the $\sin()$ function. The same goes for the other two trigonometric functions.
The process of finding the missing angle of a right angled triangle is very similar to finding the missing side.
1. | Label the sides and angle of the triangle. |
2. | Work out what trigonometric ratio you have to use (SOH, CAH or TOA). |
3. | Write down the corresponding formula. |
4. | Substitute in your values for the sides. |
5. | Perform the appropriate inverse trigonometric function to get the angle. |
Find the value of $x$ in the diagram to 1 decimal place.
Label the triangle:
You have the opposite and adjacent here. So, you have O and A. Hence, use TOA:
$\tan(x) = \frac{opposite}{adjacent}=\frac{O}{A}$
Substitute in your values:
$\tan(x)=\frac{5}{12}$
Perform the inverse tangent function to get $x$ by itself:
If $\tan(x) = \frac{5}{12}$ then $x = \tan^{-1}(\frac{5}{12})$
$x=22.619864....$
$\underline{x=22.6 \degree \ (1 d.p.)}$
Calculating the missing angle of a right-angled triangle is very similar to calculating the missing length. However, it is important to first learn about inverse trigonometric functions.
Inverse trigonometric functions invert the trigonometric function, giving the value of the angle $x$.
Function | What it's called | examples |
$\sin^{-1}()$ | Inverse sine | If $\sin(x) = 0.5$, then $x = \sin^{-1}(0.5)$ |
$\cos^{-1}()$ | Inverse cosine | If $\cos(x) = 0.8$, then $x = \cos^{-1}(0.8)$ |
$\tan^{-1}()$ | Inverse tangent | If $\tan(x) = 0.42$, then $x = \tan^{-1}(0.42)$ |
Note: The inverse function $\sin^{-1}()$ is NOT the same as $\frac{1}{\sin()}$! It is just notation to tell us that it is the inverse of the $\sin()$ function. The same goes for the other two trigonometric functions.
The process of finding the missing angle of a right angled triangle is very similar to finding the missing side.
1. | Label the sides and angle of the triangle. |
2. | Work out what trigonometric ratio you have to use (SOH, CAH or TOA). |
3. | Write down the corresponding formula. |
4. | Substitute in your values for the sides. |
5. | Perform the appropriate inverse trigonometric function to get the angle. |
Find the value of $x$ in the diagram to 1 decimal place.
Label the triangle:
You have the opposite and adjacent here. So, you have O and A. Hence, use TOA:
$\tan(x) = \frac{opposite}{adjacent}=\frac{O}{A}$
Substitute in your values:
$\tan(x)=\frac{5}{12}$
Perform the inverse tangent function to get $x$ by itself:
If $\tan(x) = \frac{5}{12}$ then $x = \tan^{-1}(\frac{5}{12})$
$x=22.619864....$
$\underline{x=22.6 \degree \ (1 d.p.)}$
Finding missing lengths in a triangle
FAQs
Question: How do I know when to use the inverse trigonometric functions?
Answer: When you want to reverse a trigonometric function that has the angle inside it. For example, when trying to solve sin(x) = 0.6, use the inverse sine function.
Question: What do the inverse trigonometric functions do?
Answer: They help you find the angle of a triangle when using the trigonometric identities.
Theory
Exercises
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