Using Fundamental Identities

Objectives:

1.Recognize and write the fundamental trigonometric identities2.Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions

WHY???Fundamental trigonometric identities can be used to simplify trigonometric expressions, such as for the coefficient of friction.

Fundamental Trigonometric Identities

Reciprocal Identities

uu

uu

uu

uu

uu

u

tan

1cot

cos

1sec

sin

1csc

cot

1tan

sec

1cos

csc

1sin

Quotient Identities

tanusinu

cosu cot u

cosu

sinu

Fundamental Trigonometric Identities

Pythagorean Identities

sin2 u cos2 u1 1 tan2 usec2 u

1 cot2 ucsc2 u

Even/Odd Identities

sin( u) sinu cos( u) cosu tan( u) tanu

csc( u) cscu sec( u) secu cot( u) cot(u)

Fundamental Trigonometric Identities

Cofunction Identities

sin2

u

= cosu cos

2

u

sinu

tan2

u

cot u cot

2

u

tanu

sec2

u

cscu csc

2

u

secu

Example: If and Ө is in quadrant II, find each function value.

a) sec Ө

To find the value of this function, look for an identity that relates tangent and secant.

Tip: Use Pythagorean Identities.

b) sin Ө

7

c) cot ( Ө )

Example: If and Ө is in quadrant II, find each function value.

(Cont.)Tip: Use Quotient Identities. Tip: Use Reciprocal and

Negative-Angle

Identities.

2. Use the values

cos x > 0 and identities to find the values of all six trigonometric functions.

1sinx and

2

What quadrant will you use? 1st quadrant

1cscx

sinx 1

1/ 2 2

2 2sin x cos x 1 2

2 1cos x 1

2

11

4 3

4

3cosx

2

1secx

cosx 2

3 2 3

3

sinxtanx

cosx

123

2

1

3

33

1cot x

tanx 3

1 3

Using Identities to Evaluate a Function

Use the given values to evaluate the remaining trigonometric functions

(You can also draw a right triangle)

secu3

2,tanu 0

csc 5,cos 0

tan x 33

,cos x 32

Simplify an ExpressionSimplify an Expression Simplify cot x cos x + sin x. Click for answer.

x

xx

sin

coscot

xx

xxx

x

xsin

sin

cossincos

sin

cos 2

xxx

xxcsc

sin

1

sin

sincos 22

2Simplif y cos xcscx cscx

Example: Simplify

1. Factor csc x out of the expression.

2cscx cos x 1

2. Use Pythagorean identities to simplify the expression in the parentheses.

2cscx cos x 1

2 2sin x cos x 1 2 2sin x cos x 1

2cscx sin x

3. Use Reciprocal identities to simplify the expression.

2cscx sin x

21sin x

sinx

2sin xsinx

sinx

Simplifying a Trigonometric Expression

sin x cos2 x sin x

sec2 x(1 sin2 x)

tan2 x

sec2 x

Factoring Trigonometric Expressions

sec2 1-Factor the same way you would factor any quadratic.- If it helps replace the “trig” word with x

-Factor the same way you would factor

sec2 1

x 2 1

x 2 1(x 1)(x 1) so sec2 (sec 1)(sec 1)

2b. 2csc x 7cscx 6

Make it an easier problem.Let a = csc x

2a2 – 7a + 6(2a – 3)(a – 2)

Now substitute csc x for a. 2cscx 3 cscx 2

2Factor sec x 3tanx 1.

1. Use Pythagorean identities to get one trigonometric function in the expression.2 2sec x tan x 1.

2tan x 1 3tanx 1

2tan x 3tanx 2

2. Now factor.

tanx 2 tanx 1

Factoring Trigonometric Expressions

4 tan2 tan 3

csc2 x cot x 3

More Factoring

sin2 x csc2 x sin2 x

1 2cos2 x cos4 x

sec3 x sec2 x sec x 1

Adding Trigonometric Expressions (Common Denominator)

sin1 cos

cossin

sinsin

sin

1 cos

cossin

(1 cos)

(1 cos)

(sin)(sin) (cos)(1 cos)

(1 cos)(sin)

sin2 cos cos2(1 cos)(sin)

1 cos

(1 cos)(sin)

1

sin

csc

sin2 cos2 1

Adding Trigonometric Expressions

1

sec x 1 1

sec x 1

tan x sec2 x

tan x

Rewriting a Trigonometric Expression so it is not in Fractional Form

1

1 sin x

5

tan x sec x

tan2 x

csc x 1

Trigonometric Substitution

4 x 2

x 2tan

64 16x 2

x 2cos

x 2 4

x 2sec

x 2 100

x 10tan