Download File - Tramore maths circle

5th Year Higher Level Maths
1.
Trigonometry Test
24/1/’14
(i) Describe how you would construct an angle of size 1 radian in a circle.
(ii) Convert 70° into radians.
(iii) Convert
5
radians into degrees.
3
2.
Solve the equation tan Ɵ = 1,
for 0° ≤ Ɵ ≤ 360°
(Ɵ in degrees)
3.
3
Solve the equation sin 2Ɵ =
,
2
for 0° ≤ Ɵ ≤ 360°
(Ɵ in degrees)
4.
Solve the equation cos 3Ɵ =
for Ɵ ϵ R
(Ɵ in radians)
5.
State the period and range of the curve below.
3
,
2
Solutions
1. (i) Draw an arc the same length as the radius and join both ends of the arc
to the centre of the circle. The angle these two radii make at the centre is one
radian in measure.
70
5
( )  1.22radians
( ) 180° = 300°
(ii)
(iii)
80
3
2.
tan is positive in 1st and 3rd quadrants.
First quadrant:
Ɵ = tan-1(1) = 45°
this is the reference angle.
rd
The other angle is in the 3 quadrant : 180° + 45° = 225°
Answer: 45° and 225°
3.
sin is positive in 1st and 2nd quadrants.
3
) = 60°
2
1st quadrant: 2Ɵ = sin-1(
this is the reference angle.
2nd quadrant: 2Ɵ = 180° - 60° = 120°
From 1st quadrant: 2Ɵ = 60°, (60° + 360°), (60° + 720°)….
From 2nd quadrant: 2Ɵ = 120°, (120° + 360°), (120° + 720°)….
From 1st quadrant: Ɵ = 30°, (30° + 180°), (30° + 360°)….
From 2nd quadrant: Ɵ = 60°, (60° + 180°), (60° + 360°)….
Angles in range:
4.
Answer = 30°, 60°, 210°, 240°
cos is positive in 1st and 4th quadrants.

3
)=
6
2
1st quadrant: 3Ɵ = cos-1(
4th quadrant: 3Ɵ =
2
-
 11
=
6
6
From 1st quadrant: 3Ɵ =

6
From 2nd quadrant: 3Ɵ =
11
6
5.
period = 120°
this is the reference angle.

Ɵ=

18
+ n 2 
Ɵ=
11
18
+ n 2
range = [-3,3]
+n
2
3
+n
2
3