9.1 Trigonometry II
• The unit circle is the circle with radius 1 unit and its centre at the origin.
0°∘
90°∘
180°∘
270°∘
,360°∘
1
45o
12
45o
sin 45°∘ cos 45°∘ tan 45°∘ 1/√2 1/√2 1
1
32 2
30o
60o
√3√32
1
Rules of Triangles sin 30°∘ cos 30°∘ tan 30°∘1/2 √3/2 1/√3
sin 60°∘ cos 60°∘ tan 60°∘√3/2 1/2 √3/2
All +Sin+
Tan + Cos +
sin θ= y 1cosθ = x 1
tanθ= y x
y
x
“CAST”
ADD SUGAR TO COFFEE
Corresponding Angle
90°∘<x<180°∘ 180°∘<x<270°∘ 270°∘<x<360°∘
θ=180 - x θ= x - 180 360 - x
θ θθ θ
ExampleDetermine the corresponding angle of the following
120°∘ 180°∘ - 120°∘ = 60°∘
200°∘ 200°∘ - 180°∘ = 20°∘
345°∘ 360°∘ - 345°∘ = 15°∘
Values of sinθ , cosθ and tanθ (0°∘ ≤ θ ≤ 360 °∘)
• sin 120°∘ = +sin ( 180°∘ - 120°∘)
= sin 60°∘
= 0.8660
sin 245°∘ = - sin(245°∘-180°∘)
= - sin 65°∘
= -0.9063
AS
T C
+++
+
Values of sinθ , cosθ and tanθ (0°∘ ≤ θ ≤ 360 °∘)
• sin 355°∘= -sin(360 - 345°∘)
= -sin15°∘=0.2588
• cos 145°∘= - cos(180°∘ -145°∘)
= -cos 35°∘=0.8192
• cos 215°∘= - cos (215°∘-180°∘)
= - cos 35°∘=0.8192
Values of sinθ , cosθ and tanθ (0°∘ ≤ θ ≤ 360 °∘)
• tan 120°∘= - tan 60°∘= 1.7321
• tan 225°∘=tan 45°∘= 1
• tan 300°∘= -tan 60°∘=-1.7321
9.1 Exercise
• Determine the values of the following
(a) sin 32°∘ (d) sin 153°∘ (g) sin 220°∘(j) sin 342°∘
(b)cos 75°∘ (e) cos 165°∘ (h) cos 268°∘ (k)cos 355°∘
(c)tan 60°∘ (f) tan 176°∘ (i) tan 245°∘ (l) tan 278°∘
P(0.5,0.8)Q(-0.9,0.6)
∙∙
∙S(0.65,-0.76)R(-0.4, -0.95)∙
The diagram shows a unit circle. Four points P, Q, R, S are marked on he circumference of the unit circle. Determinethe values of sin, cos, and tan of the following angle
(a)∠POX (b) ∠QOX (c) ∠ROX (d) reflex∠SOX
0x
• Determine whether the value of each of the following is positive or negative
(a) sin 240°∘
(b)sin 150°∘
(c)sin 335°∘
(d) cos 287°∘
(e) cos 145°∘
(f) cos 254.3°∘
(g) tan 56°∘
(h) tan 295°∘
(i) tan 278.5°∘
• Find the value of the following.
(a) 6 sin 30°∘ - 45 cos 90°∘ + 2 tan 180°∘
(b)10 cos 0°∘ X 5 sin 90°∘
(c) 6 cos 60°∘ - 2 tan 45°∘
(d) 4 sin 270°∘ x sin 90°∘ - 3 sin 180°∘
(e)4 cos 90°∘ + 7 cos 360°∘
(f) 2 tan 45°∘ + 4 tan 180°∘ + 7 tan 360°∘
• Find the value of each of the following.
(a) 3 sin 218°∘ - 4 cos 136°∘
(b)4 tan 236°∘ - 2 sin 341°∘
(c)cos 184°∘ + 2 tan 256°∘
(d)sin 225°∘ - 3 tan 348°∘
• Given that 0≤θ≤360°∘, find the values of θ for each of the following.
(a) sin θ= -0.8290
(b)sin θ= 0.2765
(c)cos θ= - 0.5646
(d)cos θ= 0.7963
(e)tan θ= -0.3547
(f) tan θ= 3.456
10 cm
33°∘
A
EDCBIn the diagram, BCDE is a straight line. Calculate(a) the length of AC(b) the length of CD(c) the length of AB(d) the value of cos ∠ADE
θ°∘
BC
DA
13 cm
12 cm11 cm
In the diagram, ABC is a straight line. Calculate the values of(a) tan∠ BDC (b)cos ∠ABD (c)sin ∠CAD (d) tan θ°∘
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