TRIGONOMETRY
Learning Outcomes
At the end of the lesson, students are able to:
• state the six trigonometric functions
• tabulate and draw graphs of sin, cos and tan functions
• define amplitude and period
• sketch graphs of y = a sin (bx) + c
y = a cos (bx) + c
y = a tan (bx) + c
1. TRIGONOMETRIC FUNCTIONS
a) The basic trig. functions are sin𝜃, cos 𝜃 and tan 𝜃.
Other trig. functions are secant 𝜃, cosecant 𝜃 and cotangent 𝜃,
where sec 𝜃
1
=
cos 𝜃
, cosec 𝜃
1
=
sin 𝜃
and cot 𝜃
1
=
tan 𝜃
b) Trigonometric ratios of special angles
𝜃
sin 𝜃
0°
0
30°
1
2
cos 𝜃
1
3
2
1
tan 𝜃 =
sin 𝜃
cos 𝜃
0
3
45°
60°
2
2
2
2
1
3
2
1
2
3
90°
1
0
undefined
c) Trigonometric ratios of complementary angles
sin 𝜃 = cos (90° - 𝜃)
cos 𝜃 = sin (90° - 𝜃)
tan 𝜃 = cot (90° - 𝜃)
d) Signs of trigonometric ratios in the four quadrants
Eg. Express the following, in terms of the basic angle :
(i) sin 210°
(iii) sin (-45°)
(ii) cos 300°
(iv) tan (- 300°)
Note: (i) Trigonometric ratios of supplementary angles
sin (180o - 𝜃) = sin 𝜃
cos (180o - 𝜃) = - cos 𝜃
tan (180o - 𝜃) = - tan 𝜃
(ii) Trigonometric ratios of negative angles
sin (- 𝜃) = - sin 𝜃
cos (- 𝜃) = cos 𝜃
tan (- 𝜃) = - tan 𝜃
e) Graphs of Trigonometric Functions
(i) y = sin𝜃, y = cos 𝜃
Complete the table below for values of sin 𝜃 and cos 𝜃, for
0o ≤ 𝜃 ≤ 360o, at 30o intervals:
Using a scale of 1 cm to represent 30o on the horizontal axis
and 1 cm to represent 0.5 on the vertical axis, draw the graphs
of y = sin 𝜃 and y = cos 𝜃, on the same diagram.
For y = sin𝜃 and
y = cos 𝜃
- Amplitude =
- Period =
Note:
- Amplitude is the distance of the min/max point from
the axis of the curve
- Period is the angle of one cycle
(ii) y = tan 𝜃
Amplitude =
Period =
Eg. Sketch the graphs of the following trig. functions, for 0o ≤ 𝜃 ≤ 360o,
stating the amplitude and the period
(i) y = sin 2𝜃
(ii) y = 3 cos 𝜃
(iii)
𝜃
y = sin
2
Eg. Sketch the graph of y = 2 cos 𝜃 + 1, for 0 ≤ 𝜃 ≤ 2𝜋, stating the amplitude
and the period
Note: y = a sin (bx) + c ⇒
y = a cos (bx) + c ⇒
y = a tan (bx) + c ⇒
Amplitude =
Period =
Amplitude =
Period =
Eg.
TRIGONOMETRIC
IDENTITIES
Learning Outcomes
At the end of the lesson, students are able to:
• state and prove the three basic trigonometric identities
• use the three basic trigonometric identities to prove other
identities
2. TRIGONOMETRIC IDENTITIES
a) The three basic trigonometric identities are :
sin2 𝜃 + cos2 𝜃 = 1
1 + tan2 𝜃 = sec2 𝜃
1 + cot2 𝜃 = cosec2 𝜃
b) Proving other Trigonometric Identities
Eg. Prove the following identities:
(i)
(ii)
TRIGONOMETRIC
EQUATIONS
Learning Outcomes
At the end of the lesson, students are able to:
• solve basic trigonometric equations
• solve other trigonometric equations
3. TRIGONOMETRIC EQUATIONS
a) Solving basic trigonometric equations
A basic trigonometric equation is of the form:
sin a𝜃 = k
cos a𝜃 = k
tan a𝜃 = k , where a and k = constants
Eg. Find all the angles x, where 0o < x < 360o , such that:
(i) sin x = 0.866
(ii) tan x = 1.732
(iii) cos x = - 0.71
(iv) sin x = cos 65o
Exercise (do in Add Math Journal)
Solve, for 0o < x < 360o :
b) Solving other trigonometric equations
Eg. Find all the angles between 0o and 360o inclusive which satisfy the
equation:
(i) 5 cos x + 2 sin x = 0
(ii) 2cos2 x – cos x = 1
(iii) 2 cos2x + 3 sin x = 3
(iv) 2 tan x = 4 – sec2 x
(v) 3 sin 2x + 2 = 0
(vi) cos (2x – 40o ) = 0.8
Eg. Solve for 0 ˂ x ˂ 2𝜋,
(ii) 2 sin (x + o.6) = - 1
(ii) tan (x –
𝜋
)
3
= √3