Trigonometry in context · Year 10
In Trigonometry in context (Year 10) you practise 6 skills: Sine, cosine and tangent · A side from an angle · An angle from two sides · Angles of elevation and depression · Two angles, two triangles · Shapes built from right triangles. In a right triangle, each acute angle has three ratios of sides. The sine is the side opposite the angle divided by the hypotenuse, the cosine is the side next to the angle divided by the hypotenuse, and the tangent is the side opposite divided by the side next to it. The ratios depend only on the angle, not on the size of the triangle: in an enlarged triangle every side grows by the same factor.
To find a side, name the known side and the side asked for, as seen from the angle, choose the ratio that links them, and work it out with the calculator’s sin, cos or tan key, set to degrees. To find an angle from two sides, divide them and use the inverse key of the same ratio, such as sin⁻¹. Round only at the end, as the question asks. An angle of elevation is measured up from the horizontal to what is seen, and an angle of depression down from the horizontal. The angle of depression from the top of a cliff to a boat equals the angle of elevation from the boat to the top, since the horizontal at the top is parallel to the ground. When there are two angles, look for two right triangles that share a side, such as the height of a tower, and find what each one gives. Other shapes are solved by cutting them into right triangles. The height to the base of an isosceles triangle, the diagonals of a rhombus and the heights of an isosceles trapezium all make right triangles, and a side found in one triangle feeds the next step, such as an area, a cost or a time.
Tip: Before choosing a ratio, mark the angle and name each side from it: opposite it, next to it, or the hypotenuse.
What you learn
Sine, cosine and tangent
For example: In the right triangle in the picture, the lengths are in centimetres. What is the cosine of the marked angle? Write the answer as a decimal.
Answer and solution
The answer: 0.8
The cosine is the adjacent side over the hypotenuse: cos α = 20 ÷ 25 = 0.8
A side from an angle
For example: A ladder 4 m long leans on a wall, at 79° to the ground. How high up the wall does the ladder reach? Round your answer to 2 decimal places.
Answer and solution
The answer: 3.93 m
The side opposite the angle is the hypotenuse times the sine of the angle: x = 4 × sin 79° ≈ 3.93 Rounded, the answer is 3.93 m.
An angle from two sides
For example: A ball is kicked from the ground 13 m from a goal and passes just over the crossbar, 244 cm high, that is 2.44 m. At what angle to the ground does the ball rise? Round your answer to the nearest degree.
Answer and solution
The answer: 11 °
The known sides are opposite the angle and next to it, so use the tangent: tan α = 2.44 ÷ 13 Find the angle on a calculator: α ≈ 11° Rounded, the angle is 11°.
Angles of elevation and depression
For example: From a point on the ground 55 m from a building, the angle of elevation to its top is 49°. How tall is the building? Round your answer to 2 decimal places.
Answer and solution
The answer: 63.27 m
The height is opposite the angle and the distance is next to it, so use the tangent: x = 55 × tan 49° ≈ 63.27 Rounded, the answer is 63.27 m.
Two angles, two triangles
For example: From the top of a tower 24 m tall, two cars are seen one behind the other on a straight road that starts at its foot. The angle of depression to the nearer car is 50°, and to the further one 21°. How far apart are the two cars? Round your answer to 2 decimal places.
Answer and solution
The answer: 42.38 m
Each car’s distance from the foot is the height divided by the tangent of its angle. The further car: 24 ÷ tan 21° ≈ 62.5221 The nearer car: 24 ÷ tan 50° ≈ 20.1384 The gap between the cars is the difference: 62.5221 − 20.1384 ≈ 42.38 Rounded, the answer is 42.38 m.
Shapes built from right triangles
For example: A rhombus has sides of 16 cm, and one of its angles is 40°. How long is the longer diagonal? Round your answer to 2 decimal places.
Answer and solution
The answer: 30.07 cm
The diagonal cuts the angle in half, so each triangle has an angle of 20°. Half the longer diagonal is next to that angle: 16 × cos 20° ≈ 15.0351 The diagonal is twice that: 2 × 15.0351 ≈ 30.07 Rounded, the answer is 30.07 cm.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).