Year 11 Extended · Unit 2 Trigonometric functions and any triangle · Lesson 1 of 6

A right-angled triangle has no 150-degree angle, so what could sin 150° mean?

Turn a radius of length 1 about its centre. The height of its tip is the sine, the sideways distance is the cosine, and both keep making sense after the angle passes 90 degrees.

Geometry and TrigonometryNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 11, Unit 1: write down the exact value of sin 45°.
    Answer√2 ÷ 2, which is about 0.71.
  2. From Year 8: which quadrant of the coordinate plane has negative x and positive y?
    AnswerThe second quadrant, top left.
  3. From Year 9: a right-angled triangle has hypotenuse 1 and an acute angle of 60°. How long is the side next to the 60° angle?
    Answer0.5. That side is cos 60° × 1.
Words used in this lesson
unit circle
a circle of radius 1 with its centre at the origin. For example: the circle the crank traces.
quadrant
one of the four quarters of the coordinate plane. For example: the second quadrant is top left.
reference angle
the acute angle between the radius and the horizontal axis. For example: the reference angle of 100° is 80°.
anticlockwise
the direction opposite to a clock's hands, in which angles are measured. For example: from the positive x-axis upward.
undefined
has no value because it needs a division by zero. For example: tan at 270°.

A pedal that keeps turning

A bicycle pedal crank in Hanoi is 1 unit long and starts pointing straight ahead, level with the axle. It then turns anticlockwise, and we agree to call the height of the pedal above the axle sin of the angle turned and its distance ahead of the axle cos of that angle. For a small angle this is the right-angled triangle you already know.

Predict first: when the crank has turned through more than a quarter of a turn, say 150°, is the pedal above or below the axle, and ahead of it or behind it? What does that make the signs of its sine and cosine? Turn the radius in the first figure to check, then use the second figure to find the other angles that give the same height.

15
0.26sin θ
0.97cos θ

θ = 15°: the tip is 0.97 across and 0.26 up. sin θ = 0.26, cos θ = 0.97, tan θ = 0.27. 1st quadrant, reference angle 15°.

Show me the steps
  1. θ = 15°: the tip is 0.97 across and 0.26 up. Both are positive, and tan θ = 0.26 ÷ 0.97 = 0.27. This is the right-angled triangle you already know, with hypotenuse 1.
  2. θ = 150°: the tip has crossed to the left of the vertical axis, so cos θ = −0.87, but it is still above the horizontal axis, so sin θ = 0.5. The ratio is negative: tan θ = −0.58. The reference angle is 180° − 150° = 30°.
  3. θ = 225°: the tip is below and to the left, so sin θ = −0.71 and cos θ = −0.71. A negative divided by a negative is positive: tan θ = 1. The reference angle is 225° − 180° = 45°.

For any angle θ, measured anticlockwise from the positive x-axis, the radius of the circle of radius 1 ends at the point (cos θ, sin θ).

  • Why it matches Year 9: for an acute angle the radius is the hypotenuse of a right-angled triangle of hypotenuse 1, so the side across is cos θ ÷ 1 and the side up is sin θ ÷ 1.
  • Signs: first quadrant, everything positive. Second quadrant (90° to 180°), only sin is positive. Third (180° to 270°), only tan is positive. Fourth (270° to 360°), only cos is positive.
  • Tangent: tan θ = sin θ ÷ cos θ, so it needs cos θ not 0.
  • Reference angle: the acute angle between the radius and the x-axis. It gives the size of the value; the quadrant gives the sign.
20
20, 160, 200, 340the four angles
0.34the size of sin (the same at all four)

r = 20°: the angles are 20°, 160°, 200° and 340°. The size of the sine is 0.34 at all four; only its sign changes.

Show me the steps
  1. r = 20°: the angles are 20°, 160°, 200° and 340°. The sine has the same size, 0.34, at all four. It is positive above the horizontal axis (20° and 160°) and negative below it.
  2. r = 50°: the angles are 50°, 130°, 230° and 310°. Reflect in the vertical axis and you go from 50° to 180° − 50° = 130°. Reflect in the horizontal axis and you go from 50° to 360° − 50° = 310°.
  3. r = 75°: the angles are 75°, 105°, 255° and 285°. The two angles above the axis, 75° and 105°, add to 180°; the two below, 255° and 285°, add to 540°, which is 180° + 360°.

The reference angle gives the size, never the sign. cos 100° is negative even though 100° is only 10° past a right angle, because the tip is to the left of the vertical axis. And the sine of an angle in the second quadrant is positive: it is the height, and the tip is still above the axis. Decide the sign from the quadrant before you write anything down.

Worked examples

Find the exact value of sin 135°.

  1. 135° is in the second quadrant, where sin is positive. The reference angle is 180° − 135° = 45°.
  2. sin 135° = sin 45° = √2 ÷ 2.

Find the exact value of cos 210°.

  1. 210° is in the third quadrant, where cos is negative. The reference angle is 210° − 180° = 30°.
  2. cos 210° = −cos 30° = −√3 ÷ 2.

Your go. The first step is given. Find the exact value of tan 300°.

  1. 300° is in the fourth quadrant, where only cos is positive, so tan is negative.
  2. Find the reference angle, then use the exact value of the tangent of it.
Check the answer

The reference angle is 360° − 300° = 60°, and tan 60° = √3. In the fourth quadrant tan is negative, so tan 300° = −√3. A common slip is to write +√3 from the acute angle.

Check before you go on

1. The tip of the radius is at an angle of 248°. Which statement about sin 248° and cos 248° is correct?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. cos 68° is about 0.37. Using the reference angle, find cos 112°, to 2 decimal places.

3. sin 23° is about 0.39. Find the other angle between 0° and 360° whose sine is also about 0.39.

Level 2 · Building

4. tan 35° is about 0.70. Find the other angle between 0° and 360° whose tangent is also about 0.70.

5. The tip of the radius is at the point (−0.6, 0.8) on the unit circle. Find tan θ, to 2 decimal places.

Level 3 · Stretch

6. The tip of a crank 18 cm long has turned anticlockwise through 240° from the horizontal, starting pointing forward. How far above the axle is the tip? Give a negative number if it is below, to 1 decimal place.

7. How many angles between 0° and 360° have a sine of 0.8?

Something to think about

Why can tan 90° not be given a value? Look at what happens to tan θ = sin θ ÷ cos θ as θ gets closer and closer to 90° from below.

One way to think about it

As θ approaches 90°, cos θ gets closer to 0 while sin θ stays near 1, so the ratio grows without limit. At 90° exactly the division would be by 0, so tan 90° is undefined, and the graph of the tangent (lesson 2) has a break there.

Exit ticket

8. For one angle between 0° and 360°, sin θ is negative and tan θ is negative. Which of these could it be?