Higher
hsn.uk.net
Mathematics
Trigonometry
Contents
Trigonometry 1
1 Radians EF 1
2 Exact Values EF 1
3 Solving Trigonometric Equations RC 2
4 Trigonometry in Three Dimensions EF 5
5 Compound Angles EF 8
6 Double-Angle Formulae EF 11
7 Further Trigonometric Equations RC 12
8 Expressing pcosx + qsinx in the form kcos(x – a) EF 14
9 Expressing pcosx + qsinx in other forms EF 15
10 Multiple Angles EF 16
11 Maximum and Minimum Values EF 17
12 Solving Equations RC 18
13 Sketching Graphs of y = pcosx + qsinx EF 20
CfE Edition
This document was produced specially for the HSN.uk.net website, and we require that any
copies or derivative works attribute the work to Higher Still Notes.
For more details about the copyright on these notes, please see
http://creativecommons.org/licenses/by-nc-sa/2.5/scotland/
, Higher Mathematics Trigonometry
Trigonometry
1 Radians EF
Degrees are not the only units used to measure angles. The radian (RAD on
the calculator) is a measurement also used.
Degrees and radians bear the relationship:
π radians
= 180°.
The other equivalences that you should become familiar with are:
30° = π6 radians 45° = π4 radians 60° = π3 radians
90° = π2 radians 135° = 34π radians 360° =2π radians.
Converting between degrees and radians is straightforward.
• To convert from degrees to radians, multiply by π
and divide by 180. × π
180
Degrees Radians
• To convert from radians to degrees, multiply by 180
× 180
π
and divide by π .
π = 5 π radians .
For example, 50°= 50 × 180 18
2 Exact Values EF
The following exact values must be known. You can do this by either
memorising the two triangles involved, or memorising the table.
DEG RAD sin x cos x tan x
0 0 0 1 0
π 1 3 1 Tip
30 6 2 2 3
You’ll probably find it
2 π 1 1 easier to remember the
45 4 2 2 1
1 triangles.
π 3 1
60 3 2 2 3
1 1
π
90 2 1 0 –
hsn.uk.net Page 1 CfE Edition
, Higher Mathematics Trigonometry
3 Solving Trigonometric Equations RC
You should already be familiar with solving some trigonometric equations.
EXAMPLES
1. Solve sin x° = 12 for 0 < x < 360 .
sin x° = 12 180° − x ° x°
S A Since sin x° is
180° + x °
T C 360° − x °
Remember
positive The exact value triangle:
First quadrant solution:
2 30° 3
x = sin −1 ( 12 )
= 30. 60°
=x 30 or 180 − 30 1
x = 30 or 150.
2. Solve cos x° = − 1 for 0 < x < 360 .
5
cos x° = − 1 180° − x ° x°
5 S A Since is
cos x°
T C 360° − x °
180° + x °
negative
x = cos −1 1 ( 5)
= 63·435 (to 3 d.p.).
x= 180 − 63·435 or 180 + 63·435
x = 116·565 or 243·435.
3. Solve sin x° =3 for 0 < x < 360 .
There are no solutions since −1 ≤ sin x ° ≤ 1 .
Note that −1 ≤ cos x ° ≤ 1 , so cos x° =3 also has no solutions.
hsn.uk.net Page 2 CfE Edition
hsn.uk.net
Mathematics
Trigonometry
Contents
Trigonometry 1
1 Radians EF 1
2 Exact Values EF 1
3 Solving Trigonometric Equations RC 2
4 Trigonometry in Three Dimensions EF 5
5 Compound Angles EF 8
6 Double-Angle Formulae EF 11
7 Further Trigonometric Equations RC 12
8 Expressing pcosx + qsinx in the form kcos(x – a) EF 14
9 Expressing pcosx + qsinx in other forms EF 15
10 Multiple Angles EF 16
11 Maximum and Minimum Values EF 17
12 Solving Equations RC 18
13 Sketching Graphs of y = pcosx + qsinx EF 20
CfE Edition
This document was produced specially for the HSN.uk.net website, and we require that any
copies or derivative works attribute the work to Higher Still Notes.
For more details about the copyright on these notes, please see
http://creativecommons.org/licenses/by-nc-sa/2.5/scotland/
, Higher Mathematics Trigonometry
Trigonometry
1 Radians EF
Degrees are not the only units used to measure angles. The radian (RAD on
the calculator) is a measurement also used.
Degrees and radians bear the relationship:
π radians
= 180°.
The other equivalences that you should become familiar with are:
30° = π6 radians 45° = π4 radians 60° = π3 radians
90° = π2 radians 135° = 34π radians 360° =2π radians.
Converting between degrees and radians is straightforward.
• To convert from degrees to radians, multiply by π
and divide by 180. × π
180
Degrees Radians
• To convert from radians to degrees, multiply by 180
× 180
π
and divide by π .
π = 5 π radians .
For example, 50°= 50 × 180 18
2 Exact Values EF
The following exact values must be known. You can do this by either
memorising the two triangles involved, or memorising the table.
DEG RAD sin x cos x tan x
0 0 0 1 0
π 1 3 1 Tip
30 6 2 2 3
You’ll probably find it
2 π 1 1 easier to remember the
45 4 2 2 1
1 triangles.
π 3 1
60 3 2 2 3
1 1
π
90 2 1 0 –
hsn.uk.net Page 1 CfE Edition
, Higher Mathematics Trigonometry
3 Solving Trigonometric Equations RC
You should already be familiar with solving some trigonometric equations.
EXAMPLES
1. Solve sin x° = 12 for 0 < x < 360 .
sin x° = 12 180° − x ° x°
S A Since sin x° is
180° + x °
T C 360° − x °
Remember
positive The exact value triangle:
First quadrant solution:
2 30° 3
x = sin −1 ( 12 )
= 30. 60°
=x 30 or 180 − 30 1
x = 30 or 150.
2. Solve cos x° = − 1 for 0 < x < 360 .
5
cos x° = − 1 180° − x ° x°
5 S A Since is
cos x°
T C 360° − x °
180° + x °
negative
x = cos −1 1 ( 5)
= 63·435 (to 3 d.p.).
x= 180 − 63·435 or 180 + 63·435
x = 116·565 or 243·435.
3. Solve sin x° =3 for 0 < x < 360 .
There are no solutions since −1 ≤ sin x ° ≤ 1 .
Note that −1 ≤ cos x ° ≤ 1 , so cos x° =3 also has no solutions.
hsn.uk.net Page 2 CfE Edition
