Analytic Trigonometry

1. TOPIC 1.2 – ANALYTIC
TRIGONOMETRY
1.2.1: The Inverse Sine, Cosine, and Tangent Functions
1.2.2: The Inverse Trigonometric Functions
1.2.3: Trigonometric Identities
1.2.4: Sum and Difference Formulas
1.2.5: Double-Angle and Half-Angle Formulas
1.2.6: Product-to-Sum and Sum-to-Product Formulas
1.2.7: Trigonometric Equations (I)
1.2.8: Trigonometric Equations (II)
1

2. 2
Review of Properties of Functions and Their Inverses
1.2.1& 1.2.2: The Inverse
Sine, Cosine, and Tangent
Functions

3. 3

4. 1
sin
xy sin
22
x
y x x ysin sin1
The inverse sine function:
The inverse sine function denoted by
sine function from . Thus,
is the inverse of the restricted
4

5. x1
sin
x1
sin
Finding exact values of
1. Let =
xsin2. Rewrite = asx1
sin
xsin3. Use the exact values to find the value of that satisfies
2
3
sin 1
2
2
sin 1
Example: Find the exact value of;
1- 2-
5

6. 6

7. 1
cos
xy cos x0
The inverse cosine function:
The inverse cosine function denoted by
restricted cosine function from . Thus,
is the inverse of the
y x x ycos cos1
0 y 1 1xwhere and
Example: Find the exact value of;
1)
2
1
cos 1
7

8. 8

9. 1
tan
xy tan
2 2
y
The inverse tangent function:
The inverse tangent function denoted by
restricted tangent function from .Thus,
is the inverse of the
y x x ytan tan1
2 2
y xwhere and
Example: Find the exact
value of;
1) 1tan 1
9

10. Composition of functions involving
inverse trigonometric functions
xx1
sinsin
xxsinsin 1
2
,
2
Inverse properties
1.Sine function:
for every x in the interval [-1,1]
for every x in the interval
xx1
coscos
xxcoscos 1
,0
2. Cosine function:
for every x in the interval [-1,1]
for every x in the interval
xx1
tantan
xxtantan 1
2
,
2
3. Tangent function:
for every real number x
for every x in the interval
10

11. 7.0coscos 1 sinsin 1
2coscos 1
Example:
1-Find the exact value if possible;
a) b) c)
4
3
tansin 1
2
1
sincos 1d) e)
2- If x > 0, write x1
tansec as an algebraic expression in x
11

12. 1.2.3: Trigonometric Identities
Fundamental trigonometric identities
i ii iii.csc
sin
.sec
cos
.cot
tan
1 1 1
Reciprocal Identities
i ii.tan
sin
cos
.cot
cos
sin
Quotient Identities
i ii
iii
.sin cos .tan sec
. cot csc
2 2 2 2
2 2
1 1
1
Pythagorean Identities
i ii iii
iv v vi
.sin( ) sin .cos( ) cos .tan( ) tan
.csc( ) csc .sec( ) sec .cot( ) cot
Even - Odd Identities
12

13. Example: Verify the identity:
Changing to sine and cosine
1)
xxx sectancsc 2) xxxx cscsincotcos
3) xx
xx
xx
cossin
cscsec
)csc(sec
Using factoring
1)
xxxx 32
sincossinsin 2) x
x
x
x
x
csc2
sin
cos1
cos1
sin
Multiplying numerator and denominator by the same factor
1)
x
x
x
x
cos
sin1
sin1
cos
Working with both sides separately
1) 2
tan22
sin1
1
sin1
1
13

14. 1.2.4: Sum and Difference
Formulas
Sum and difference formulas for cosines and sines
cos (A+ B) = cos A cos B - sin A sin B
cos (A - B) = cos A cos B + sin A sin B
sin (A + B) = sin A cos B + cos A sin B
sin (A - B) = sin A cos B - cos A sin B
Example:
1.Using difference formula to find the exact value
a) Give the exact value of

6090cos30cos using the sum and difference formula
12
5
sin
4612
5
b) Find the exact value of using the fact that
14

15. tantan1
coscos
cos
5
4
sin angle
2
1
sin
angle
2. Verify the identity:
3. Suppose that for a quadrant II and
for a quadrant I . Find the exact value of
cos cos
cos sin
a) b)
c) d)
15

16. tan( )
tan tan
tan tan1
tan( )
tan tan
tan tan1
Sum and difference formulas for tangents
1-
2-
Example:
1.Verify the identity: xx tantan
16

17. 1.2.5: Double-Angle and
Half-Angle Formulas
cossin22sin
22
sincos2cos
2
tan1
tan2
2tan
Double angle formulas:
1-
2-
3-
5
4
sin
Example:
1- If and lies in quadrant II, find the exact value of;
2sin 2cos 2tana) b) c)
2. Find the exact value of 15sin15cos 22
17

18. Using Pythagorean identity to write 2cos in terms of sine only:
22
sincos2cos
1cos22cos 2
2
sin212cos
Three forms of the double angle formula for cos
1-
2-
3-
Example: Verify the identity: 3
sin4sin33sin
2
2cos1
sin2
2
2cos1
cos2
2cos1
2cos1
tan2
Power reducing formulas
x4
sin
of trigonometric functions greater than 1
that does not contain powersExample: Write an equivalent expression for
18

19. Half angle formulas
cos1
cos1
2
tan;
2
cos1
2
cos;
2
cos1
2
sin
The + or – in each formula is determined by the quadrant in which
2
lies
Example:
1- Use cos 120º to find the exact value of cos 105º
2- Verify the identity:
2cos1
2sin
tan
2
sin
cos1
2
tan
cos1
sin
2
tan
Half angle formula for tan
Example:
Verify the identity: csccscsec
sec
2
tan
19

20. 1.2.6: Product-to-Sum and
Sum-to-Product Formulas
)]cos()[cos(
2
1
sinsin
)]cos()[cos(
2
1
coscos
)]sin()[sin(
2
1
cossin
)]sin()[sin(
2
1
sincos
1-
2-
3-
4-
Example: Express each of the following products as a sum or difference:
a. sin 5x sin 2x b. cos 7x cos x
20

21. 2
cos
2
sin2sinsin
2
cos
2
sin2sinsin
2
cos
2
cos2coscos
2
sin
2
sin2coscos
Sum to Product Formulas:
1-
2-
3-
4-
Example:
1- Express each sum or difference as a product
a. sin 7x + sin 3x b. cos 3x +cos 2x
2- Verify the identity: x
xx
xx
tan
sin3sin
cos3cos
21

22. 1.2.7& 1.2.8: Trigonometric
Equations
• A trigonometric equation is an equation that contains a trigonometric expression.
• To solve an equation containing a single trigonometric function:
 Isolate the function on one side of the equation
 Solve for the variable
Finding all solutions of a trigonometric equation
Example: Solve the equation: 3sin3sin5 xx
Solving an equation with a multiple angle
Example: Solve the equation:
32tan x
20 x
2
1
3
sin
x
20 x
1- 2-
22

23. 01sin3sin2 2
xx 20 x
xxx sintansin 20 x
Trigonometric equations quadratic in form
Example: Solve the equation:
Using factoring to separate 2 different trigonometric functions in an equation
Example: Solve the equation:
0sin2cos xx 20 x
2
1
cossin xx 20 x
1sincos xx 20 x
Using an identity to solve a trigonometric equation
Example: Solve the equation:
1-
2-
3-
23