5.1 Using Fundamental Identities
What are we learning?
-Recognize and write the fundamental identities
-Use the fundamental identities to evaluate, simplify,
and rewrite trigonometric expressions.
Why are we learning it?
Identities can be used to simplify expressions, making
problems easier to solve. Plus, it’s fun!
What is an identity?
Reciprocal identities
sin u =
cos u =
tan u =
csc u =
sec u =
cot u =
Quotient identities
tan u =
Pythagorean Identities
sin2u + cos2u = 1
cot u =
Cofunction Identities
sin( – u) =
tan( – u) =
sec( – u) =
cos( – u) =
cot( – u) =
csc( – u) =
Even/Odd Identities
sin(-u) =
cos(-u) =
tan(-u) =
csc(-u) =
sec(-u) =
cot(-u) =
Example 1: Use sin x = ½ and cos x > 0 to find values of all six trigonometric functions.
Example 2: Simplify cos2x csc x – csc x.
Example 3: Factor.
a) 1 – cos2x
c) Factor sec2x + 3tanx + 1
b) 2csc2x – 7csc x + 6
Example 4: Simplify csc t – cos t cot t.
Example 5: Add and simplify
Example 6: Rewrite
.
so it is not in fractional form.
Example 7: Use the substitution x = 5sinθ, 0 < θ < π/2 to express √
.
Example 8: Rewrite ln|secθ| + ln|cotθ|as a single logarithm and simplify the result.
5.2 Verifying Trigonometric Identities
What are we learning?
-Verify trigonometric identities.
Why are we learning it?
We can use identities to rewrite problems that model
real life situations, often making them easier to solve.
How to do it:
1)
2)
3)
4)
5)
Work with one side of the equation at a time (usually the more complicated side.)
Try to factor, and add fractions, square a binomial, or create a monomial denominator.
Try to use the fundamental identities.
If all else fails, change everything to sine and cosine.
Keep trying, even if you make mistakes. Practice makes perfect!
Example 1: Verify the identity.
Example 2: Verify the identity.
Example 3: Verify (sec2x - 1)(sin2x – 1) = -sin2x
Example 4: cscx – sinx = cosxcotx
Example 5: cscβ + cotβ =
Example 6:
Example 7: tan3x = tanxsec2x – tanx
5.3 Solving Trigonometric Equations
What are we learning?
-Using algebra to solve trigonometric equations.
-Solve trigonometric equations involving multiple
angles.
Why are we learning it?
You can use trigonometric equations to model real
world situations.
Example 1: Solve for x.
2sinx = 1
Example 2: Solve sinx - √ = -sinx
Example 3: Solve 4sin2x – 3 = 0
Example 4: Solve sin2x = 2sinx
Example 5: Find all solutions of 2sin2x – 3sinx + 1 = 0 in the interval [0,2π].
5.4 Sum and Difference Formulas
What are we learning?
-Use sum and difference formulas to evaluate, verify,
and simplify trigonometric expressions
Why are we learning it?
-Trigonometric functions can be used in many real
world situations.
More identities!
sin(u + v) =
sin(u – v) =
cos(u + v) =
cos(u – v) =
tan(u + v) =
tan(u – v) =
When will we use these?
Example 1: Find the exact value.
a) sin75⁰
c) cos25⁰cos20⁰ - sin25⁰sin20⁰
b) cos
Example 2: Find the exact values of the sine, cosine, and tangent of the angle by using a sum or difference.
Example 3: Simplify.
a) sin(
b) tan(
Example 4: Write sin(arctan1 +arccosx) as an algebraic expression.
Example 5: Prove the cofunction identity sin(
5.5 Multiple Angle and Product to Sum Formulas
What are we learning?
-Use multiple angle, power-reducing, half angle, and
product to sum formulas to evaluate, verify, and
simplify trigonometric expressions
Why are we learning it?
-Trigonometric functions can be used in many real
world situations.
Even more identities!
Double-angle formulas
sin2u =
cos2u =
Example 1: Solve cos2x + cosx = 0
Example 2: Use sinu = 3/5 0 < u < π/2 to find sin2u, cos2u, and tan2u.
Example 3: Derive a triple angle formula for cos3x.
tan2u =
Power-reducing formulas
sin2u =
cos2u =
tan2u =
Example 4: Rewrite tan4x as a quotient of first powers of the cosines of multiple angles.
Half-angle
sin =
cos =
tan =
Example 5: Find the value of cos105⁰