Trigonometry (lecture)
The three major trigonometric ratios:
𝑎
sin 𝜃 =
𝑐
𝑏
cos 𝜃 =
𝑐
𝑎
tan 𝜃 =
𝑏
Exact values of some trigonometric ratios
General Angles
The plane is divided into 4 quadrants by the axes.
All angles are measured from the positive 𝑥-axis (initial
line).
Anti-clocwise angles are +𝑖𝑣𝑒 and clockwise angles −𝑖𝑣𝑒.
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The basic angle, 𝛂 , is the smallest angle made with the
horizontal axis (+𝑥-axis or −𝑥-axis).
The basic angle is an unsigned angle, thus for mathematical
calculations its taken as positive.
sin(90 − θ) = cos θ
cos(90 − θ) = sin θ
1
tan(90 − θ) =
tan θ
cos(−𝜃) = cos 𝜃
sin(−𝜃) = − sin 𝜃
tan(−𝜃) = − tan 𝜃
cos20
Q1. Evaluate without using the calculator,
cos20+sin70
Q2. (N14/11/Q1)
Trigonometric ratios and the four quadrants
Given one trig ratio, the other trig ratios can be calculated by
making a right-angled triangle with lengths according to the
value of the ratio, and in the respective quadrant in
accordance with the sign of the ratio.
Q3. Given that tan 𝜃 = −1/2 and that tan 𝜃 and sin 𝜃 have opposite
signs, find the value of cos 𝜃 and sin 𝜃.
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Q4. Given that sin 𝐴 = − where, 270° < 𝐴 < 360°, find the value of
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cos 𝐴 , tan 𝐴 and of cos(−𝐴).
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Q5. (J14/12/Q3)
Trigonometric Equations
Basic trig ratios of the form, sin 𝜃 = 𝑘 , cos 𝜃 = 𝑘 ,
tan 𝜃 = 𝑘, where 𝑘 is a constant, are solved as follows:
Step1: Find the basic angle 𝛼.
Step2: Considering the sign of 𝑘, map 𝛼 in the possible
quadrant(s).
Step3: Measuring from the initial line, with reference to the basic
angles marked, find all the values of 𝜃 in the required interval.
Q6. Find all the values of 𝜃 such that
a. cos 𝜃 = 0.46 , where −360° < 𝜃 < 360°.
b. sin 𝜃 = −0.951 , where 90° < 𝜃 < 270°.
c. tan 𝜃 = 4 , where −2𝜋 < 𝜃 < 2𝜋.
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d. tan 𝜃 = −√ , two smallest positive values of 𝜃 in radians.
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e. sin 𝜃 = 0, where −360° ≤ 𝜃 ≤ 360°.
f. sin 𝜃 = 1, where −360° ≤ 𝜃 ≤ 360°.
g. sin 𝜃 = −1, where −360° ≤ 𝜃 ≤ 360°.
h. cos 𝜃 = 0, where −360° ≤ 𝜃 ≤ 360°.
i. cos 𝜃 = 1, where −360° ≤ 𝜃 ≤ 360°.
j. cos 𝜃 = −1, where −360° ≤ 𝜃 ≤ 360°.
k. tan 𝜃 = 0, where −360° ≤ 𝜃 ≤ 360°.
In case of multiple or compound angles first alter the range
(step 0)
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Q7. Find the value(s) of 𝑥.
𝑥
a. sin 2 = 0.8480 , where −360° ≤ 𝑥 ≤ 360°
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b. cos 4𝑥 = − 4 , where −360° ≤ 𝑥 ≤ 360°
c. tan(2𝑥 + 60) = −0.3584 , where −360° ≤ 𝑥 ≤ 360°
𝑥 𝜋
d. √2 sin (3 + 4 ) = 1 , where 0 < 𝑥 < 4π
WORKSHEET 1
sin 𝜃
tan 𝜃 ≡ cos 𝜃
sin2 𝜃 + cos2 𝜃 ≡ 1
o sin2 𝜃 ≡ 1 − cos2 𝜃
o cos2 𝜃 ≡ 1 − sin2 𝜃
When solving to find multiple solutions, never cancel a trig
ratio, always separate it as common.
Look out for disguised quadratics; powers in the ratio 2: 1
Using the identities try to bring down the equation in terms
of one trig ratio.
Q8. Find all the angles between -360° and 360° inclusive for which
a. 4 sin2 𝜃 cos 𝜃 = tan2 𝜃
b. 2 cos 3 𝜃 − 5 cos 2 𝜃 − 3 cos 𝜃 = 0
Q9. Find all the angles between −2𝜋 and 2𝜋 rad which satisfy the
equation
a. cos 2 𝑥 − 2 sin2 𝑥 = 0
b. sin2 𝑥 +3 sin 𝑥 cos 𝑥 + 2 cos 2 𝑥 = 0
Q10. (N14/11/Q3)
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Q11. (N12/12/Q6)
WORKSHEET 2
Proving trigonometric identities
The following guidelines should prove helpful.
o It is easier to break higher powers into lower ones.
o It is easier to merge two terms into one rather than breaking
a single term into two.
o If both sides are basic forms then forcefully transform one
side by multiplying and dividing with the same single
expression (take hints from the other end).
o 𝐬𝐢𝐧𝟐 𝜽 ≡ 𝟏 − 𝐜𝐨𝐬 𝟐 𝜽 ≡ (𝟏 − 𝐜𝐨𝐬 𝜽)(𝟏 + 𝐜𝐨𝐬 𝜽)
𝐜𝐨𝐬 𝟐 𝜽 ≡ 𝟏 − 𝐬𝐢𝐧𝟐 𝜽 ≡ (𝟏 − 𝐬𝐢𝐧 𝜽)(𝟏 + 𝐬𝐢𝐧 𝜽)
o Sometimes replacing a 1 by 𝐬𝐢𝐧𝟐 𝜽 + 𝐜𝐨𝐬 𝟐 𝜽 helps.
Q12. Prove the following trigonometric identities:
a. sin4 θ − cos 4 θ ≡ 1 − 2 cos 2 θ
b. tan2 θ − sin2 θ ≡ tan2 θ sin2 θ
1+sin θ cos θ
c. ≡
cos θ 1−sin θ
d. (1 − sin 𝜃 + cos 𝜃)2 ≡ 2(1 − sin 𝜃)(1 + cos 𝜃)
1+sin θ 1 2
e. ≡ (tan 𝑥 + )
1−sin θ cos 𝑥
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Q13. (J14/13/Q4)
WORKSHEET 3
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