Worksheet 3.3 Trigonometry
Section 1 Review of Trig Ratios
Worksheet 2.8 introduces the trig ratios of sine, cosine, and tangent. To review the ratios,
consider a triangle ABC with angle φ as marked.
B
c a
φ
b
A C
The hypotenuse (hyp) of the triangle is c; the adjacent (adj) side is b; the opposite (opp) side
is a. The side of length a is opposite the angle, and the side of length b is the side adjacent to
the angle which is not the hypotenuse. Then we have
opp a
sin φ = =
hyp c
adj b
cos φ = =
hyp c
opp a
tan φ = =
adj b
Note also that a
sin φ c a
= b
= = tan φ
cos φ c
b
Exercises:
1. For the following triangle, find the ratios:
Z
(a) sin θ Z
Z
(b) tan θ 3 Z 5
Z
(c) cos θ Z
θ ZZ
4
1
, 2. For the following triangle, find the ratios:
a
Z
(a) tan θ (d) sin φ Zθ
Z
(b) cos φ (e) tan φ Z b
cZ
(c) sin θ (f) cos θ Z φ
Z
Z
3. (a) Use Pythagoras’ theorem to find x
H
(b) Find (i) sin θ HH
H
(ii) tan θ 5 H x
HH
(iii) cos θ θ HH
H
12
Section 2 Degrees and Radians
Recall from Worksheet 2.9 that
π
1◦ = radians
180
In university maths it is much more common to give angles in radians rather than degrees.
If the units are left off an angle, then the angle is in radians. Degrees can be converted to
radians using the above formula, but it will be very convenient for you to know some standard
conversions. In particular:
π
90◦ = 2
30◦ = π6
π
45◦ = 4
180◦ = π
π
60◦ = 3
360◦ = 2π
Example 1 : An equilateral 4 has three equal angles of π3 .
Example 2 : Convert 50◦ to radians.
π 5π
50◦ = 50 = radians
180 18
π π 180◦
Example 3 : How many degrees is 9
radians? We know 180◦ = π, so 9
= 9
therefore
π
= 20◦
9
2
Section 1 Review of Trig Ratios
Worksheet 2.8 introduces the trig ratios of sine, cosine, and tangent. To review the ratios,
consider a triangle ABC with angle φ as marked.
B
c a
φ
b
A C
The hypotenuse (hyp) of the triangle is c; the adjacent (adj) side is b; the opposite (opp) side
is a. The side of length a is opposite the angle, and the side of length b is the side adjacent to
the angle which is not the hypotenuse. Then we have
opp a
sin φ = =
hyp c
adj b
cos φ = =
hyp c
opp a
tan φ = =
adj b
Note also that a
sin φ c a
= b
= = tan φ
cos φ c
b
Exercises:
1. For the following triangle, find the ratios:
Z
(a) sin θ Z
Z
(b) tan θ 3 Z 5
Z
(c) cos θ Z
θ ZZ
4
1
, 2. For the following triangle, find the ratios:
a
Z
(a) tan θ (d) sin φ Zθ
Z
(b) cos φ (e) tan φ Z b
cZ
(c) sin θ (f) cos θ Z φ
Z
Z
3. (a) Use Pythagoras’ theorem to find x
H
(b) Find (i) sin θ HH
H
(ii) tan θ 5 H x
HH
(iii) cos θ θ HH
H
12
Section 2 Degrees and Radians
Recall from Worksheet 2.9 that
π
1◦ = radians
180
In university maths it is much more common to give angles in radians rather than degrees.
If the units are left off an angle, then the angle is in radians. Degrees can be converted to
radians using the above formula, but it will be very convenient for you to know some standard
conversions. In particular:
π
90◦ = 2
30◦ = π6
π
45◦ = 4
180◦ = π
π
60◦ = 3
360◦ = 2π
Example 1 : An equilateral 4 has three equal angles of π3 .
Example 2 : Convert 50◦ to radians.
π 5π
50◦ = 50 = radians
180 18
π π 180◦
Example 3 : How many degrees is 9
radians? We know 180◦ = π, so 9
= 9
therefore
π
= 20◦
9
2
