Academy Algebra IIPre-Calculus (5.1, 5.2)

14.3: Trigonometric IdentitiesHW: today: none,

HW tomorrow: p.928 (16-34 even)

Trig Identities• Reciprocal Identities

• Quotient Identities

uu

uu

uu

uu

uu

uu

tan

1cot

cos

1sec

sin

1csc

cot

1tan

sec

1cos

csc

1sin

u

uu

u

uu

sin

coscot

cos

sintan

Trig Identities• Pythagorean Identities

• Even/Odd Identities

uu

uu

uu

22

22

22

csccot1

sectan1

1cossin

uuuu

uuuu

uuuu

cot)cot( tan)tan(

sec)sec( cos)cos(

csc)csc( sin)sin(

Trig Identities• Cofunction Identities

uuuu

uuuu

uuuu

sec2

csc csc2

sec

tan2

cot cot2

tan

sin2

cos cos2

sin

Simplify the Trig Expressionxxx sincossin 2

Simplify the Trig Expression))(secsin1( 2 xx Check using your calculator:

Y1= Question

Y2= Answer

Check: graphs/tables are the same

Verify.

xxx

xtansin

cos2

cos2

Verify the Identity Algebraically. 22 sincosseccos

Verify the Identity Algebraically.

seccoscot

csc1

Factor the expression and simplify.xxx 222 sectansec

5.2: Verifying Trig Identities1.) Work with one side of the equation.2.) Algebra: Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator.3.) Trig: Look for opportunities to use the fundamental identities.4.) If the preceding guidelines do not help, try converting to all sines and cosines.5.) Try something! Even making an attempt that leads to a dead end gives insight.

Verify.1csc

1csc

1csc2

xx

x

Multiply and simplify. xxxx csccotcsccot

Verify.x

xx2cot2

1sec

1

1sec

1

Verify the Identity.

22

2

sinsec

1sec

Verify the Identity.

2sec2

sin1

1

sin1

1

Verify the Identity. xxx 222 tan1cos1tan

Verify the Identity.xxxx cscseccottan

Verify the Identity.

x

xxx

sin1

costansec

Verify the Identity.

sin

sin1

csc1

cot 2