term 3 : unit 1 trigonometry (part b) name : ____________ ( ) class : ______ date :________ 1.3...
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Term 3 : Unit 1Trigonometry (Part B)
Name : ____________ ( ) Class : ______ Date :________
1.3 Simple Identities
1.4 Trigonometric Equations
Simple Trigonometric Identities and Equations
1.3 Simple Identities
In this lesson, we will
• define the secant, cosecant and cotangent functions,
• learn some simple trigonometric identities.
Objectives
Trigonometric Ratios of Acute AnglesThe three
trigonometric ratios are defined
as
OPQ is a right angled triangle.
adjacent
oppositehypotenuse
cosec
sec
cot
oppositeoppositehypotenusehypotenuse
adjacentadjacent
1
sin
1
cos
1
tan
Simple Trigonometric Identities and Equations
O
y
x
P (x, y)
r
Consider angles in the Cartesian plane.
Simple Trigonometric Identities and Equations
tany
x
yrxr
sintan
cos
2 22 2sin cos
y x
r r
2 2
2
y x
r
2
2
r
r
2 2sin cos 1 For any value
of θ.
r2 = x2 + y2
Simple Trigonometric Identities and Equations
1cot
tan
sin
cos
1
coscot
sin
2 2sin cos 1 2 2sin cos 1 2 2
2 2 2
sin cos 1
cos cos cos
2 2
2 2 2
sin cos 1
sin sin sin
2 2tan 1 sec 2 21 cot cosec
Simple Trigonometric Identities and Equations
Given that 2sin and cos 1, show thatx y 22 4 1 4x y
2 2sin cos 1
22 4 1 4x y
2sin sin2
xx
cos 1 cos 1y y
2
21 1
2
xy
From the identity
Rearranging
Rearranging
Example 3
Simple Trigonometric Identities and Equations
21 cosec
Simplify .1 sin 1 sin
x
x x
2cosec x
21 cosec
1 sin 1 sin
x
x x
2
2
cot
1 sin
x
x
2
1
sin x
2
2
cossin
cos
xxx
Rearranging 1 + cot2x = cosec2x
Using the identities
Cancelling
Example 1
Simple Trigonometric Identities and Equations
Prove the following identity sec cos sin tan .x x x x
sin tanx x
sec cosx x 1cos
cosx
x
2sin
cos
x
x
21 cos
cos
x
x
Using the identity
sinsin
cos
xx
x
Example 2
Simple Trigonometric Identities and Equations
Given that cosec cot 3, evaluate cosec cot and cos .A A A A A
cosec cot cosec cot 3 cosec cotA A A A A A
Using the identity
1 + cot2x = cosec2x.
cosec cot 3A A
2 2cosec cot 3 cosec cotA A A A 1 3 cosec cotA A
13cosec cotA A
13cosec cot cosec cot 3A A A A
1032cosecA 53cosecA
1sin
cosecA
A
2351
35
452cos 1 sinA A
Example 3
Simple Trigonometric Identities and Equations
2 2Show that sin can be expressed as cot and
1 cos 1 cos
find the value of .
k
k
2 2sin
1 cos 1 cos
4cos
sin
2
2 2cos 2 2cossin
sin
Using the identity.4cot
2
2 1 cos 2 1 cossin
1 cos
Using the identity.
Example 4
4k
Simple Trigonometric Identities and Equations
1.4 Trigonometric Equations
In this lesson, we will solve some further trigonometric equations by
simplifying or factorising, to reduce them to the form sin x = k, cos x = k and
tan x = k.
Objectives
O
y
x
AS
T C
x
Find all the angles between 0° and 360° which satisfy the equation 3 cos x + 2 sin x = 0.
Simple Trigonometric Identities and Equations
3cos 2sin 0x x
2sin 3cosx xsin 3
cos 2
x
x
56.3
123.7 , 303.7x , 0180 36x
cos x ≠ 0
3tan
2x
3tan
2
tan x < 0 so x is in the 2nd or the 4th quadrant.
Using the identity.
Calculate the base angle α.
Example 5
O
y
x
AS
T C
y
Find all the angles between 0o and 360o which satisfy the equation sin y = 4 tan y.
Simple Trigonometric Identities and Equations
sin 4 tan , 0 360o oy y y sin
sin 4cos
yy
y
180oy sin 0y
sin cos 4siny y ysin cos 4sin 0y y y
Using the identity
Factorise, do not cancel
through by sin θ. sin cos 4 0y y
cos 4y No solutions –1 ≤ θ ≤ 1
Example 6
O
y
x
AS
T C
y
= 30°
Find all the angles between 0° and 360° which satisfy the equation 2 cos2 y – 1 = sin y.
Simple Trigonometric Identities and Equations
22cos 1 sin , 0 360y y y 22 1 sin 1 siny y
270y , 130 50y
Using sin2y + cos2y =
1
22 2sin 1 siny y
sin y > 0 so y is in the 1st or the 2nd quadrant.
Factorising
22sin sin 1 0y y 2sin 1 sin 1 0y y
12sin y sin 1y
O
y
x
AS
T C
y
Example 7
O
y
x
AS
T C
x + 30°
Find all the angles between 0° and 360° which satisfy the equation cos (x + 30o) = – 0.3.
Simple Trigonometric Identities and Equations
cos 30 0.3x
0 360x 30 30 390x
72.5
30 107.5 , 252.5x
30 18 , 1800x cos 0.3
cos (x + 30°) < 0 so x is in the 2nd or the 3rd quadrant. Calculate the
basic angle α.
72.5 , 18030 180 72.5x
77.5 , 222.5x
Example 8
O
y
x
AS
T C
2x
Find all the angles between 0° and 360° which satisfy the equation sin 2x = 0.866.
Simple Trigonometric Identities and Equations
sin 2 0.866x
0 360x 0 2 720x
60
20 , 4260 , 1 0 , 480
2 , 360, , 540180x sin 0.866
sin 2x > 0 so x is in the 1st or the 2nd quadrant. Calculate the
basic angle α.
, 366 0 , 5400 , 180 60 60 60
30 , 60 , 210 , 240x
Example 9
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