Trigonometric ratios Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Trigonometric ratios questions. See exactly how to solve problems on finding a missing side, SOHCAHTOA, exact trigonometric values, special angles.

finding a missing sideSOHCAHTOAexact trigonometric valuesspecial anglesrounding to a given accuracyfinding a missing angle
GCSE Foundation70 questionsStep-by-step solutions
Question 1
2 markseasy
A right-angled triangle has an acute angle of 30∘30^\circ. The hypotenuse has length 1212 cm. Work out the length of the side opposite this angle. Give your answer as an exact value.

Worked solution

  1. Label the sides of the triangle with respect to the given angle.

    hypotenuse=12 cm,opposite=x cm\text{hypotenuse} = 12\text{ cm}, \quad \text{opposite} = x\text{ cm}

    The hypotenuse is the side opposite the right angle. Looking from the 30∘30^\circ angle, the opposite side is the one across the triangle from it and the adjacent side is the one beside it. Here you are given the hypotenuse and asked for the opposite.

  2. Choose the ratio that links the side you know to the side you want.

    sin⁡θ=opphyp\sin \theta = \frac{\text{opp}}{\text{hyp}}

    The hypotenuse and the opposite are exactly the two sides in the sine ratio (SOH), so use sin⁡\sin. Neither of the other two ratios uses both of these sides.

  3. State the length of the opposite side.

    x=12×12=6 cmx = 12 \times \frac{1}{2} = 6\text{ cm}

    The opposite side is 66 cm long.

Answer
x=6 cmx = 6\text{ cm}
Question 2
2 markseasy
A right-angled triangle has an acute angle of 60∘60^\circ. The hypotenuse has length 1414 cm. Work out the length of the side adjacent to this angle. Give your answer as an exact value.

Worked solution

  1. Label the sides of the triangle with respect to the given angle.

    hypotenuse=14 cm,adjacent=x cm\text{hypotenuse} = 14\text{ cm}, \quad \text{adjacent} = x\text{ cm}

    The hypotenuse is the side opposite the right angle. Looking from the 60∘60^\circ angle, the opposite side is the one across the triangle from it and the adjacent side is the one beside it. Here you are given the hypotenuse and asked for the adjacent.

  2. Choose the ratio that links the side you know to the side you want.

    cos⁡θ=adjhyp\cos \theta = \frac{\text{adj}}{\text{hyp}}

    The hypotenuse and the adjacent are exactly the two sides in the cosine ratio (CAH), so use cos⁡\cos. Neither of the other two ratios uses both of these sides.

  3. State the length of the adjacent side.

    x=14×12=7 cmx = 14 \times \frac{1}{2} = 7\text{ cm}

    The adjacent side is 77 cm long.

Answer
x=7 cmx = 7\text{ cm}
Question 3
2 markseasy
A right-angled triangle has an acute angle of 45∘45^\circ. The side adjacent to this angle has length 99 cm. Work out the length of the side opposite this angle. Give your answer as an exact value.

Worked solution

  1. Label the sides of the triangle with respect to the given angle.

    adjacent=9 cm,opposite=x cm\text{adjacent} = 9\text{ cm}, \quad \text{opposite} = x\text{ cm}

    The hypotenuse is the side opposite the right angle. Looking from the 45∘45^\circ angle, the opposite side is the one across the triangle from it and the adjacent side is the one beside it. Here you are given the adjacent and asked for the opposite.

  2. Choose the ratio that links the side you know to the side you want.

    tan⁡θ=oppadj\tan \theta = \frac{\text{opp}}{\text{adj}}

    The adjacent and the opposite are exactly the two sides in the tangent ratio (TOA), so use tan⁡\tan. Neither of the other two ratios uses both of these sides.

  3. State the length of the opposite side.

    x=9×1=9 cmx = 9 \times 1 = 9\text{ cm}

    The opposite side is 99 cm long.

Answer
x=9 cmx = 9\text{ cm}
Question 4
2 markseasy
A right-angled triangle has an acute angle of 30∘30^\circ. The side opposite this angle has length 77 cm. Work out the length of the hypotenuse. Give your answer as an exact value.

Worked solution

  1. Label the sides of the triangle with respect to the given angle.

    opposite=7 cm,hypotenuse=x cm\text{opposite} = 7\text{ cm}, \quad \text{hypotenuse} = x\text{ cm}

    The hypotenuse is the side opposite the right angle. Looking from the 30∘30^\circ angle, the opposite side is the one across the triangle from it and the adjacent side is the one beside it. Here you are given the opposite and asked for the hypotenuse.

  2. Choose the ratio that links the side you know to the side you want.

    sin⁡θ=opphyp\sin \theta = \frac{\text{opp}}{\text{hyp}}

    The opposite and the hypotenuse are exactly the two sides in the sine ratio (SOH), so use sin⁡\sin. Neither of the other two ratios uses both of these sides.

  3. State the length of the hypotenuse side.

    x=712=14 cmx = \frac{7}{\frac{1}{2}} = 14\text{ cm}

    The hypotenuse side is 1414 cm long.

Answer
x=14 cmx = 14\text{ cm}
Question 5
2 markseasy
A right-angled triangle has an acute angle of 35∘35^\circ. The hypotenuse has length 1212 cm. Work out the length of the side opposite this angle. Give your answer correct to 11 decimal place.

Worked solution

  1. Label the sides of the triangle with respect to the given angle.

    hypotenuse=12 cm,opposite=x cm\text{hypotenuse} = 12\text{ cm}, \quad \text{opposite} = x\text{ cm}

    The hypotenuse is the side opposite the right angle. Looking from the 35∘35^\circ angle, the opposite side is the one across the triangle from it and the adjacent side is the one beside it. Here you are given the hypotenuse and asked for the opposite.

  2. Choose the ratio that links the side you know to the side you want.

    sin⁡θ=opphyp\sin \theta = \frac{\text{opp}}{\text{hyp}}

    The hypotenuse and the opposite are exactly the two sides in the sine ratio (SOH), so use sin⁡\sin. Neither of the other two ratios uses both of these sides.

  3. State the length of the opposite side.

    x=12×sin⁡35∘=6.8829…=6.9 cmx = 12 \times \sin 35^\circ = 6.8829\ldots = 6.9\text{ cm}

    The opposite side is 6.96.9 cm long.

Answer
x=6.9 cmx = 6.9\text{ cm}

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