Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Lesson 1: which two quadrants have a positive sine?
Answer
The first and second quadrants. - From Lesson 2: what is the period of y = tan x?
Answer
180°. - From Year 9: solve 3x + 5 = 20.
Answer
x = 5. 3x = 15.
Words used in this lesson
- solution
- a value of the unknown that makes the equation true. For example: an angle that gives the right sine.
- interval
- all the values between two end values. For example: 0° to 360°.
- symmetry
- a graph that looks the same after a reflection. For example: the sine graph about 90°.
- inverse
- the operation that undoes another. For example: sin⁻¹ undoes sin.
- principal value
- the one angle that the calculator's inverse key gives. For example: −30° for sin⁻¹ (−0.5).
The seat on the wheel
A seat travels once round a wheel in Osaka. On the way up it passes the 12 m mark; on the way down it passes the 12 m mark again. The height depends on the angle turned, and the angles for the two passes are different.
Predict first: in one complete turn, how many times is the seat exactly 12 m high? And if the first time is at an angle of 40°, how would you find the other without any calculation? Use the first figure to test your idea before reading the key idea.
sin x = 0.2: the solutions from 0° to 360° are 11.5° and 168.5°. There are 2.
Show me the steps
- k = 0.2: the calculator gives 11.5°. The curve is symmetric about 90°, so the other crossing is 180° − 11.5° = 168.5°. There are 2 solutions.
- k = −0.6: the calculator gives −36.9°. The crossings are below the axis, at 180° + 36.9° = 216.9° and 360° − 36.9° = 323.1°. Again 2 solutions.
- k = 0: the line is the axis itself, and it meets the curve at 0°, 180° and 360°. That is 3 solutions, because both ends of the interval count.
Solving for 0° ≤ x ≤ 360°. Find the angle p the calculator gives for the inverse, then use the symmetry of the graph for the rest.
- sin x = k: the solutions are p and 180° − p. The sine is the height, and the circle is symmetric in the vertical axis.
- cos x = k: the solutions are p and 360° − p. The cosine is the sideways distance, and the circle is symmetric in the horizontal axis.
- tan x = k: the solutions are p and p + 180°, because the tangent repeats every 180°.
- Rearrange first: isolate sin x, cos x or tan x. If an equation has sin x and cos x as a ratio, sin x ÷ cos x = tan x.
- Check the interval: a negative p means add 360° (or find the quadrants); a solution outside the interval is thrown away.
tan x = 1.5: the solutions from 0° to 360° are 56.3° and 236.3°. There are 2.
Show me the steps
- k = 1.5: the calculator gives 56.3°. The tangent repeats every 180°, so the next solution is 56.3° + 180° = 236.3°. There are 2.
- k = −2.5: the calculator gives −68.2°, outside the interval. Adding 180° and 360° gives 111.8° and 291.8°, 180° apart again.
- k = 0: the solutions are where the curve crosses the axis, 0°, 180° and 360°. The tangent has no solution at 90° or 270°, where it is undefined.
One calculator key does not give every solution. After the first angle, ask which other quadrant has the same sign for this ratio. Then sketch the graph or the circle and check that every answer has the right sign.
Worked examples
Solve 2 sin x = 1.4 for 0° ≤ x ≤ 360°.
- Divide by 2: sin x = 0.7. The calculator gives p = 44.4°.
- Sine is positive in the first and second quadrants: x = 44.4° and x = 180° − 44.4° = 135.6°.
Solve cos x = −0.2 for 0° ≤ x ≤ 360°.
- The calculator gives p = 101.5°, which is already in the second quadrant, where cosine is negative.
- The other solution is in the third quadrant: x = 360° − 101.5° = 258.5°. So x = 101.5° or 258.5°.
Your go. The first step is given. Solve tan x = 2 for 0° ≤ x ≤ 360°.
- The calculator gives p = tan⁻¹ 2, in the first quadrant.
- The tangent repeats every 180°. Add 180° to p for the second solution.
Check the answer
p = 63.4°, and the second solution is 63.4° + 180° = 243.4°. So x = 63.4° or 243.4°. A common slip is to use 180° − 63.4° = 116.6°, which has a tangent of −2.
With a calculator
Put the calculator in degree mode first: sin 30 should give 0.5. The inverse keys, labelled sin⁻¹, cos⁻¹ and tan⁻¹ or arcsin, arccos and arctan, give one angle only. For example sin⁻¹ 0.9 shows 64.2, and the second solution 180 − 64.2 is yours to find.
When the value is negative, the key for sine or tangent shows a negative angle, such as −64.2 for sin⁻¹(−0.9). That is outside 0 to 360, so use it as the reference angle and place it in the correct quadrants, or add 360 to bring it into the interval.
Check before you go on
1. Solve cos x = 0.3 for 0° ≤ x ≤ 360°. The calculator shows 72.5°. Which gives all the solutions?
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. sin x = 0.85 for 0° ≤ x ≤ 360°, and the calculator gives 58.2°. Find the other solution, to 1 decimal place.
3. Solve tan x = 1.2 for 0° ≤ x ≤ 360°. Give the larger solution, to 1 decimal place.
Level 2 · Building
4. Solve 4 sin x = −1 for 0° ≤ x ≤ 360°. Give the smaller solution, to 1 decimal place.
5. Solve 2 cos x + 1 = 0 for 0° ≤ x ≤ 360°. Give the larger solution in degrees.
Level 3 · Stretch
6. Solve 3 sin x = 2 cos x for 0° ≤ x ≤ 360°. Give the larger solution, to 1 decimal place.
7. How many solutions does cos x = 0.25 have for 0° ≤ x ≤ 720°?
Something to think about
For which values of k does sin x = k have no solution at all? What does the horizontal line y = k look like then, and what does the unit circle say?
One way to think about it
When k is bigger than 1 or smaller than −1. The line y = k sits above the highest point of the sine curve, or below the lowest. On the unit circle the height of the tip is never more than 1, so no angle can have a sine of 1.5.
Exit ticket
8. Solve 5 tan x + 4 = −3 for 0° ≤ x ≤ 360°. Which pair is correct, to 1 decimal place?