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°∘