Finding angles in right-angled triangles - KS3 Maths

Key points

can be used to find a missing angle in a right-angled triangle when two sides are known.
An understanding of the three trigonometric ratios and function machines is essential.
For right-angled triangles with angle \(Ɵ\), the three are:

sin\(Ɵ\) = \(\frac{opposite}{hypotenuse}\)

cos\(Ɵ\) = \(\frac{adjacent}{hypotenuse}\)

tan\(Ɵ\) = \(\frac{opposite}{adjacent}\)

To find a missing angle, two sides are into one of the trigonometric equations above. The equation used must contain the two sides that are given in the question.
To find an for \(Ɵ\), do the function of sin, cos or tan. The inverse functions of sin, cos and tan are sin⁻¹, cos⁻¹ and tan⁻¹.
To work out \(Ɵ\), calculate sin⁻¹ or cos⁻¹ or tan⁻¹ of the fraction on the right-hand side. sin⁻¹ can usually be written into a scientific calculator by pressing 'shift' then 'sin', and similarly for cos⁻¹ and tan⁻¹.
Sin, cos and tan are commonly used abbreviations for the three trigonometric functions sine, cosine and tangent.

How to calculate an unknown angle in a right-angled triangle

For a right-angled triangle, follow these steps to calculate an angle \(Ɵ\), when two sides are given:

Choose one of the three depending on which two sides are given in the diagram:

sin\(Ɵ\) = \(\frac{opposite}{hypotenuse}\)

cos\(Ɵ\) = \(\frac{adjacent}{hypotenuse}\)

tan\(Ɵ\) = \(\frac{opposite}{adjacent}\)

the two sides given in the diagram into the trigonometric equation.
To find an expression for \(Ɵ\), do the function of sin, cos or tan. The inverse function of sin is sin⁻¹. The inverse function of cos is cos⁻¹. The inverse function of tan is tan⁻¹.
To find the value of \(Ɵ\), calculate sin⁻¹ or cos⁻¹ or tan⁻¹ of the fraction on the right hand side. sin⁻¹ or cos⁻¹ or tan⁻¹ can usually be inputted on a scientific calculator by pressing shift before sin, cos or tan.

Note that scientific calculators need to be used for trigonometry and should be in degrees mode. Make sure there is a small D at the top of the calculator screen, and if not, go into the calculator settings to change the angle unit to 'degrees' (or 'deg').

Examples

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Image caption,
Calculate the value of Ɵ.
Image caption,
Label the sides given. The adjacent side (𝒂) is the side between the right angle and the angle Ɵ. Label 16 cm as 𝒂. The hypotenuse (𝒉) is the longest side and is opposite the right angle. Label 32 cm as 𝒉. There is no need to label the opposite side because there is no information about its length.
Image caption,
Choose which of the three trigonometric ratios to use. The equation has to contain 𝒂 and 𝒉 because these are the sides with information in the diagram. The only equation that contains both 𝒂 and 𝒉 is the cosine (cos) ratio.
Image caption,
Write down the trigonometric equation, cosƟ = 𝒂/𝒉. Substitute the values of 𝒂 and 𝒉 into the equation. 𝒂 is 16 and 𝒉 is 32. This gives cosƟ = 16/32
Image caption,
The aim is to find the value of Ɵ that makes cosƟ equal 16/32. A function takes an input and does something to it to create an output. An inverse function takes the output and does the opposite to get back to the input. The cosine (cos) function takes the angle Ɵ and gives the answer to adjacent ÷ hypotenuse (in this example, 16/32). cos⁻¹ is the inverse function of cos. The cos⁻¹ function takes the answer to adjacent ÷ hypotenuse (in this example, 16/32), and gives the angle Ɵ. To find Ɵ, calculate cos⁻¹ (16/32).
Image caption,
To write cos⁻¹ on a scientific calculator, press 'shift' and 'cos'. Then type 16/32 using the fraction button. Close the brackets after typing in the fraction. The value of Ɵ is 60°.

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Image caption,
Calculate the value of Ɵ to 1 dp.
Image caption,
Label the sides given. The opposite side (𝒐) is the side opposite the angle Ɵ. Label 5 cm as 𝒐. The adjacent side (𝒂) is the side between the right angle and the angle Ɵ. Label 7 cm as 𝒂. There is no need to label the hypotenuse because there is no information about its length.
Image caption,
Choose which of the three trigonometric ratios to use. The equation has to contain 𝒐 and 𝒂 because these are the sides with information in the diagram. The only equation that contains both 𝒐 and 𝒂 is the tangent (tan) ratio.
Image caption,
Write down the trigonometric equation, tanƟ = 𝒐/𝒂. Substitute the values of 𝒐 and 𝒂 into the equation. 𝒐 is 5 and 𝒂 is 7. This gives tanƟ = 5/7
Image caption,
The aim is to find the value of Ɵ that makes tanƟ equal 5/7. A function takes an input and does something to it to create an output. An inverse function takes the output and does the opposite to get back to the input. The tan function takes the angle Ɵ and gives the answer to opposite ÷ adjacent (in this example, 5/7). tan⁻¹ is the inverse function of tan. The tan⁻¹ function takes the answer to opposite ÷ adjacent (in this example, 5/7) and gives the angle Ɵ. To find Ɵ, calculate tan⁻¹ (5/7).
Image caption,
To write tan⁻¹ on a scientific calculator, press 'shift' and 'tan'. Then type 5/7 using the fraction button. Close the brackets after typing in the fraction. The value of Ɵ is 35.5376….°. Round the answer to one decimal place to give Ɵ = 35.5°.