1. Maths revision course by Miriam Hanks
Trigonometry
Trigonometry is anything using
sin, cos or tan
Maths revision course by Miriam Hanks

2. Maths revision course by Miriam Hanks
Trigonometry - Radians
What is a radian ? r r
One radian is the angle of a circle which has an arc length equal to the radius.
There are 2π radians in 360° 1c r
Maths revision course by Miriam Hanks

3. Maths revision course by Miriam Hanks
Trig – Radians to degrees
Converting:
radians
degrees
degrees
radians
Maths revision course by Miriam Hanks

4. Maths revision course by Miriam Hanks
Trigonometry – Exact values
Your exact values need to be learned:
30°
45°
60°
sin
cos
tan 1
Maths revision course by Miriam Hanks

5. Maths revision course by Miriam Hanks
Trigonometry – Graphs
y = sin x
Maths revision course by Miriam Hanks

6. Maths revision course by Miriam Hanks
Trigonometry – Graphs
y = cos x
Maths revision course by Miriam Hanks

7. Maths revision course by Miriam Hanks
Trigonometry – Graphs
y = tan x
Maths revision course by Miriam Hanks

8. Maths revision course by Miriam Hanks
Trigonometry – CAST
When do you use CAST?
Use CAST quadrants whenever you press
sin-1, or cos-1 or tan-1 on your calculator.
180 - x x S A T C
180 + x
360 - x
Maths revision course by Miriam Hanks

9. Trigonometry – formulae
Remember: sin2 x means (sinx)2
sin 2x = 2 sin x cos x
cos 2x = cos2 x – sin2 x
cos 2x = 2 cos2 x – 1
cos 2x = 1 – 2sin2 x
All these formulae are on the formula sheet
Maths revision course by Miriam Hanks

10. Trigonometry – formulae
sin (p ± q) = sin p cos q ± cos p sin q
cos (p ± q) = cos p cos q + sin p sin q
These are also on the formula sheet
Maths revision course by Miriam Hanks

11. Trigonometry – formulae
These 2 are NOT on the formula sheet
Maths revision course by Miriam Hanks

12. Maths revision course by Miriam Hanks
Solving trig equations
If there are “double angles and single angles”, get rid of the double angles by using the formulae on the sheet for sin 2A or cos 2A.
eg 3sinx – 1 = cos 2x
single angle
double angle
3sinx – 1 = 1 – 2sin2x
Maths revision course by Miriam Hanks

13. Maths revision course by Miriam Hanks
Solving trig equations
If there is a squared term, then put everything onto one side, and factorise.
3sinx – 1 = 1 – 2sin2x
squared term
2sin2x + 3sinx – 2 = 0
(2sinx – 1)(sinx + 2)= 0
sin x = ½ or sinx = -2
(impossible)
Maths revision course by Miriam Hanks

14. Maths revision course by Miriam Hanks
Solving trig equations
If there is only one of sin cos or tan, then solve by doing the inverse function:
sin x = ½ give answers 0 < x <
x = sin-1(½)
x = or √ √
(π – x) S A T C
Maths revision course by Miriam Hanks

15. Maths revision course by Miriam Hanks
Trigonometry –The wave function
If you have a sin x and a cos x and a constant, it is not possible to solve the equation using the above techniques, so we use the wave function to reduce it to a single trig function eg
sin x + 5cos x = 6 or
3sin x – 2cos x = -1
½cos x + 7sin x – 2= 0
Maths revision course by Miriam Hanks

16. Trigonometry –The wave function
eg Express
3 cos x – 4 sin x into the form k cos (x + a)
Expand
k cos (x + a) using the formula sheet first
Maths revision course by Miriam Hanks

17. Maths revision course by Miriam Hanks
Trigonometry –The wave function
k cos (x + a) = kcos x cos a - k sin x sin a
Then put this equal to the function in the question:
3 cos x – 4 sin x = kcos x cos a – k sin x sin a
Maths revision course by Miriam Hanks

18. Maths revision course by Miriam Hanks
Trigonometry –The wave function
3 cos x – 4 sin x = kcos x cos a – k sin x sin a
Now compare how many cos x ’s you have on each side of the equation:
3 = k cos a
Now compare how many sin x ’s you have on each side of the equation:
4 = k sin a
Maths revision course by Miriam Hanks

19. Maths revision course by Miriam Hanks
Trigonometry –The wave function
3 = k cos a
4 = k sin a
Use to divide these equations:
tan a = 4/3
a = tan-1 (4/3)
a = radians (to 3 dps)
Maths revision course by Miriam Hanks

20. Maths revision course by Miriam Hanks
Trigonometry –The wave function
3 = k cos a
4 = k sin a
Use Pythagoras to find k:
k =
k = 5
So, the answer is 5 cos (x )
Maths revision course by Miriam Hanks

21. Maths revision course by Miriam Hanks
Trigonometry –Max & min values
If you are asked for a maximum or minimum value and the question is only worth one point,
then use the fact that sin of anything and cos of anything have max & min values of 1 and –1.
eg What are the maximum and minimum values of y when y = 2 + 3cos(x – 40).
Answer: Since cos(x – 40) has max value 1, then the max value of y is 5. What is the min?
Maths revision course by Miriam Hanks

22. Maths revision course by Miriam Hanks
Trigonometry in real life
When is trigonometry used in real life?
Tidal experts are meteorologists use sine waves to predict tides. These are invaluable to sailors and kayakers when planning a trip.
Maths revision course by Miriam Hanks