Download Unit 7 – Trigonometric Identities and Equations 7.1 – Exploring

Unit 7 – Trigonometric Identities and Equations
7.1 – Exploring Equivalent Trig Functions
When we look at the graphs of sine, cosine, tangent and their reciprocals, it
is clear that there will be points where two or more of them intersect. This
represents a point where trig functions are equal. We will be exploring this
today.
Odd & Even Functions
i)
The Sine Function


ii)
F(x) = sin(x) is an odd function
Therefore, f(-x) = -sin(x)
The Cosine Function


iii)
F(x) = cos(x) is an even function
Therefore, f(-x) = cos(x)
The Tangent Function

F(x) = tan(x) is a(n) ______ function
How do we tell? We look at the angle (-x) which is actually 2pi – x
How are Sine and Cosine Related?
Graph the sine function, what transformation will turn it into the cosine
function?
Graph the cosine function, what transformation will turn it into the sine
function?
(Answers are:
cos(x – pi/2) = sin(x)
= sin(x + pi/2) = cos(x)
Relating a Function to Its Period
Sine Function, period = 2pi
Sin(x + 2pi) = ______________
Cosine Function, period = 2pi
Cos(x + 2pi) = ______________
Tangent Function, period = _____
Tan(x + ___) = _______________
Relationships between Complimentary Angles (Cofunction Identites)
Complimentary angles are (x) and (pi/2 – x)

The easiest way to explore the relationship is by setting up a right
angle triangle (find sine and cosine for both x, and pi/2 – x and
compare) – also find for tan(x) and tan(pi/2 – x) using sine and cosine
A
x
pi/2 - x
C
Tan (x) = sin(x)
Cos (x)
B
tan(pi/2 – x) = sin(pi/2 – x) = cos(x) = cot(x)
cos(pi/2 – x) sin(x)
Relationships between Angles in Different Quadrants

Assume (x) is the principal angle in quadrant I, how does this relate to
the related acute angles in each quadrant? Show using cart. plane
Quadrant 2 = pi – x
Quadrant 3 = pi + x
Quadrant 4 = 2pi – x
Example: Pg. 393 #5a Write an expression that is equivalent to sin(7pi/8)
using the related acute angle


Sin(7pi/8) is in the second quadrant, and is therefore positive
The only other quadrant sine is positive in is the first quadrant and the
related acute angle would be sin(pi/8)
Example: Pg. 393 #7a State whether cos(x + 2pi) = cos (x). Justify your
decision.

+
The period for cosine is 2pi, therefore x + 2pi is equal to x
7.2 – Compound Angle Formulas

An angle that is created by adding or subtracting two or more angles
When are compound angles used?




To obtain exact values for trigonometric rations (break angles into
angles from special triangles)
To show that trig ratios are equivalent
To prove trig identities (7.4)
To solve trig equations
Compound Angle Formulas
Addition Formulas
sin(𝑎 + 𝑏) = sin 𝑎 cos 𝑏 + cos 𝑎 sin 𝑏
Subtraction Formulas
sin(𝑎 − 𝑏) = sin 𝑎 cos 𝑏 − cos 𝑎 sin 𝑏
cos(𝑎 + 𝑏) = cos 𝑎 cos 𝑏 − sin 𝑎 sin 𝑏
cos(𝑎 − 𝑏) = cos 𝑎 cos 𝑏 + sin 𝑎 sin 𝑏
tan(𝑎 + 𝑏) =
tan 𝑎 + tan 𝑏
1 − tan 𝑎 tan 𝑏
tan(𝑎 + 𝑏) =
tan 𝑎 − tan 𝑏
1 + tan 𝑎 tan 𝑏
Example: Rewrite sin(2a)cos(a) + cos(2a)sin(a) as a single trigonometric
ratio.
Sin(2a + a) or sine(3a)
Example: Determine the exact value of cos(15°)
Cos(15)
=
=
=
=
cos(45 – 30)
cos(45)cos(30) + sin(45)sin(30)
(√2/2)(√3/2) + (√2/2)(1/2)
√6 + √2
4
Example: Simplify cos(7pi/12)cos(5pi/12) - sin(7pi/12)sin(5pi/12)
= cos (7pi/12 + 5pi/12)
= cos (pi)
= -1
Example: Evaluate sin(x + y) if sin x = 4/5 and sin y = -12/13, where
0 < x < pi/2, 3pi/2 < y < 2pi
Sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
Opp = 4
Hyp = 5
Adj = √(52 – 42)
= √(25 – 16)
= √9
=3
Opp = -12
Hyp = 13
Adj = √(132 – 122)
= √(169 – 144)
= √25
=5
Cos(x) = 3/5
Cos(y) = 5/13
Sin(x + y) = (4/5)(5/13) + (3/5)(-12/13)
= 20/65 – 36/65
= -16/65
Try Pg. 400 #1-4,5abc,6,8cde,9-13,15
7.3 – Double Angle Formulas


Show how a double angle (2x) relate to the original angle (x)
Can all be derived from the compound angle formulas
Example: Use the compound angle formula for sine to show that
Sin(2x) = 2sin(x)cos(x)
Sin(2x) = sin(x + x)
= sin(x)cos(x) + cos(x)sin(x)
= 2sin(x)cos(x)
Example: Use the compound angle formula to derive the double angle
formula for cosine and tangent:
Cos(2x) = cos(x + x)
= cos(x)cos(x) – sin(x)sin(x)
= cos2x – sin2x
Given that sin2x + cos2x = 1
= cos2x – (1 – cos2x)
Cos(2x) = 2cos2x – 1
OR
OR
= (1 – sin2x) – sin2x
= 1 – 2sin2x
Tan(2x) = tan(x) + tan(x)
1 – tan(x)tan(x)
= 2tan(x)
1 – tan2x
Example: Write 2sin4xcos4x as a single trigonometric ratio.
Sin(2x) = 2sin(x)cos(x)
2sin(4x)cos(4x)
in this case x = 4x
Sin(2(4x)) = 2sin(4x)cos(4x)
Sin(8x) = 2sin(4x)cos(4x)
Example: Express 1 – 2sin2(3pi/8) as a single trig ratio and evaluate
Cos(2x) = 1 – 2sin2x
in this case x = 3pi/8
Cos(2(3pi/8)) = Cos(3pi/4) with related acute angle in Q2 (pi/4)
Cos(3pi/4) = √2/2
Example: Use the double angle formula to rewrite tan6x
Tan(6x) = Tan2(3x)
= 2tan(3x)
1 – tan2(3x)
Example: Determine the values of sin2x and cos2x, given cos x = -2/3 and
0 ≤ x ≤ pi
Adj = -2 (therefore in Q2)
Hyp = 3
Opp = √(32 – 22)
= √5 (positive in Q2)
Sin(x) = √5/3
Sin(2x) = 2sin(x)cos(x)
= 2(√5/3)(-2/3)
= -4√5
9
Cos(2x) = 1 - 2sin2x
= 1 – 2(√5/3)2
= 1 – 2(5/9)
= 1 – 10/9
= -1/9
Try Pg. 407 #1-11,15a,16a,17