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Trigonometric Ratios Introduction

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Trigonometric Ratios Introduction

Use right-angled triangles to find unknown sides and angles. Label sides relative to the marked angle, choose the correct ratio, and show your working. Use degree mode on your calculator; round decimal answers to 3 significant figures unless told otherwise.

Set up the ratios

A drone survey uses right-angled triangles to estimate distances. For each question, pay attention to which angle is marked: opposite and adjacent depend on that angle, while the hypotenuse is always opposite the right angle.

1.Complete the ratio guide for angle θ: sin θ = ___ / ___; cos θ = ___ / ___; tan θ = ___ / ___. Then state the mnemonic commonly used to remember these ratios.

2.A right-angled triangle has sides 5 cm, 12 cm and 13 cm. Angle θ is opposite the 5 cm side. Find sin θ, cos θ and tan θ as simplified fractions.

3.A right-angled triangle has hypotenuse 17 cm and one shorter side 8 cm. Find the other side, then calculate sin, cos and tan of the angle opposite the 8 cm side. Give ratios as fractions.

Choose and use a ratio

A field crew in Aotearoa is estimating heights and distances without measuring directly. Sketch and label a right triangle where useful. Include units in your answers.

4.A surveyor stands 24 m from the base of a vertical pōhutukawa tree. The angle of elevation to its top is 38°. Which ratio relates the unknown height to the 24 m horizontal distance? Write an equation for the height h, then calculate h to 3 significant figures. Ignore the surveyor’s eye height.

5.A ramp rises 0.72 m over a horizontal run of 3.0 m. Find the angle the ramp makes with the horizontal, to the nearest 0.1°. State the ratio and inverse operation used.

6.A kite string is 35 m long and makes an angle of 52° with the level ground. Assuming the string is straight and taut, calculate the kite’s vertical height above the person’s hand, to 3 significant figures.

7.True or false? In any right-angled triangle, the hypotenuse is the side opposite the angle you are working with, even if that angle is not the right angle. Correct the statement if it is false.

TrueFalse

Check your reasoning

Ratios can be used to find an unknown side or an unknown angle. Decide what is known first, then select the ratio that links those sides.

8.A right-angled triangle has an angle of 29° and a hypotenuse of 14 cm. Which calculation gives the side opposite the 29° angle? Choose one and explain why: A) 14 cos 29° B) 14 sin 29° C) 14 tan 29°.

A) 14 cos 29°
B) 14 sin 29°
C) 14 tan 29°

9.From a point on level ground, the angle of elevation to the top of a lookout is 41°. The point is 18 m from the lookout’s base. A student writes sin 41° = height/18. Identify the mistake and write the correct equation.

10.A right triangle has opposite side 9 cm and adjacent side 12 cm for angle θ. Find θ to the nearest degree. Show the ratio and calculator operation.

11.A right-angled triangle has θ = 63° and adjacent side 7.5 cm. Calculate the hypotenuse to 3 significant figures. Explain why cosine is appropriate.