Page 184 - Mathematics_Form_Two

Trigonometry

4. Express each of the following in Ttherefore, the sine of an angle is equal
terms of cos 20º: to the cosine of its complement and vice
(a) cos 160° (b) cos 200° versa. From Figure 8.15, it follows that
sin A
(c) cos 340° sin A = = a a c c = = a a
÷ ÷ ÷
cosA
5. Express each of the following in cosA b b b b c c
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terms of tan 40º: (b) tan 220° But tan A = a .
Mathematics for Secondary Schools 6. Without using a calculator, find the Hence, cos A = tan A . .
c

(a) tan 140°
sin A
(c) tan 320°
value of each of the following:
Also,
and
⎛
⎞
6 ⎟
sin(−150)cos315⎜
⎟
⎜
(a)
tan300
⎜ ⎜
⎝
⎛ ⎜ 12 ⎟ ⎟ ⎟ ⎠ .
⎞
2
⎟
(b) tan(−30)cos60 ⎜ 6 ⎟ But a + 2 c = 2 b (Pythagoras’ theorem)
(Pythagoras theorem)
⎜
⎟
⎟
sin(−45) ⎜ 12 ⎟ 2
⎜ ⎜
⎠
⎝
2
2
Thus, sin A + cos A = b = 1.
b 2
Relationship between trigonometric Therefore, sin A + cos A =1.
2
2
ratios
Consider ΔABC as shown in Figure 8.15. Generally, for any angle θ, the
corresponding trigonometrical identity is
sin q 2 cos q + 2 = 1.
Example 8�15
4
sinθ
Given that sin θ = , find cos θ and
9
tan θ for 0º ≤ θ ≤ 90º.
Solution
Figure 8�15: Right-angled triangle ABC Given sin θ = 4 .
sinθ
In Figure 8.15, the angles A and C are From sin q 2 cos q + 9 2 = 1,
complementary. That is,
A + C = 90° cos 1 sinq 2 2 q = -
Thus, C = 90° − A. 16
2
But sin q = .
a a 81
But sin A = and cos C = . 16 65
b b cos q = 1- =
2
Thus, sin A = cosC cos(90= °− A). 81 65 81 65
Hence, sin A = cos(90°− A). Hence, cosθ = ± 81 = ± 9 .

178
Student's Book Form Two

11/10/2024 20:14:05
MATHEMATIC F2 v5.indd 178
MATHEMATIC F2 v5.indd 178 11/10/2024 20:14:05