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Maths

Finding missing angles in a triangle

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Finding missing angles in a triangle

In a nutshell

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.



The inverse trigonometric functions

Inverse trigonometric functions invert the trigonometric function, giving the value of the angle xx​.

Function

What it's called

examples
​sin⁡−1()\sin^{-1}()​​

Inverse sine

If sin⁡(x)=0.5\sin(x) = 0.5, then x=sin⁡−1(0.5)x = \sin^{-1}(0.5)​​
​cos⁡−1()\cos^{-1}()​​

Inverse cosine

If cos⁡(x)=0.8\cos(x) = 0.8​, then x=cos⁡−1(0.8)x = \cos^{-1}(0.8)​​​
​tan⁡−1()\tan^{-1}()​​

Inverse tangent

    If tan⁡(x)=0.42\tan(x) = 0.42​, then x=tan⁡−1(0.42)x = \tan^{-1}(0.42)​​​


Note: The inverse function sin⁡−1()\sin^{-1}() is NOT the same as 1sin⁡()\frac{1}{\sin()}​! It is just notation to tell us that it is the inverse of the sin⁡()\sin()​ function. The same goes for the other two trigonometric functions.



Calculating the missing angle

The process of finding the missing angle of a right angled triangle is very similar to finding the missing side.

PROCEDURE

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.


Example

Find the value of xx in the diagram to 1 decimal place.

Maths; Trigonometry; KS3 Year 7; Finding missing angles in a triangle


Label the triangle:

Maths; Trigonometry; KS3 Year 7; Finding missing angles in a triangle


You have the opposite and adjacent here. So, you have O and A. Hence, use TOA:

tan⁡(x)=oppositeadjacent=OA\tan(x) = \frac{opposite}{adjacent}=\frac{O}{A}


Substitute in your values:

tan⁡(x)=512\tan(x)=\frac{5}{12}​


Perform the inverse tangent function to get xx by itself:

 If tan⁡(x)=512\tan(x) = \frac{5}{12} then x=tan⁡−1(512)x = \tan^{-1}(\frac{5}{12})

​

​x=22.619864....x=22.619864....​​


​x=22.6° (1d.p.)‾\underline{x=22.6 \degree \ (1 d.p.)}


Want to find out more? Check out these other lessons!

Finding missing lengths in a triangle

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Frequently Asked Questions (FAQ)

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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