4. You can use trigonometric identities to simplify
trigonometric expressions. Recall that an
identity is a mathematical statement that is true
for all values of the variables for which the
statement is defined.
5. A derivation for a Pythagorean identity is
shown below.
x2
+ y2
= r2
cos2
θ + sin2
θ = 1
Pythagorean Theorem
Divide both sides by r2
.
Substitute cos θ for and
sin θ for
6. To prove that an equation is an identity, alter one
side of the equation until it is the same as the
other side. Justify your steps by using the
fundamental identities.
7. Example 1A: Proving Trigonometric Identities
Prove each trigonometric identity.
Choose the right-hand side
to modify.
Reciprocal identities.
Simplify.
Ratio identity.
8. Example 1B: Proving Trigonometric Identities
Prove each trigonometric identity.
1 – cot θ = 1 + cot(–θ)
= 1 + (–cotθ)
= 1 – cotθ
Choose the right-hand side
to modify.
Reciprocal identity.
Negative-angle identity.
Reciprocal identity.
Simplify.
9. You may start with either side of the given
equation. It is often easier to begin with the more
complicated side and simplify it to match the
simpler side.
Helpful Hint
10. Check It Out! Example 1a
Prove each trigonometric identity.
sin θ cot θ = cos θ
cos θ = cos θ
Choose the left-hand side
to modify.
Ratio identity.
Simplify.
cos θ
11. Check It Out! Example 1b
Prove each trigonometric identity.
1 – sec(–θ) = 1 – secθ
Choose the left-hand side
to modify.
Reciprocal identity.
Negative-angle identity.
Reciprocal Identity.
12. You can use the fundamental trigonometric
identities to simplify expressions.
If you get stuck, try converting all of the
trigonometric functions to sine and cosine
functions.
Helpful Hint
13. Example 2A: Using Trigonometric Identities to
Rewrite Trigonometric Expressions
Rewrite each expression in terms of cos θ,
and simplify.
sec θ (1 – sin2
θ)
cos θ
Substitute.
Multiply.
Simplify.
14. Example 2B: Using Trigonometric Identities to
Rewrite Trigonometric Expressions
Rewrite each expression in terms of sin θ, cos θ,
and simplify.
sinθ cosθ(tanθ + cotθ)
sin2
θ + cos2
θ
Substitute.
Multiply.
Simplify.
1 Pythagorean identity.
15. Check It Out! Example 2a
Rewrite each expression in terms of sin θ, and
simplify.
Pythagorean identity.
Simplify.
Factor the difference of two squares.
16. Check It Out! Example 2b
Rewrite each expression in terms of sin θ, and
simplify.
cot2
θ
csc2
θ – 1 Pythagorean identity.
Substitute.
Simplify.
17. Example 3: Physics Application
At what angle will a wooden block on a concrete
incline start to move if the coefficient of friction
is 0.62?
Set the expression for the weight component
equal to the expression for the force of friction.
mg sinθ = μmg cosθ
sinθ = μcosθ
sinθ = 0.62 cosθ
Divide both sides by mg.
Substitute 0.62 for μ.
18. Example 3 Continued
tanθ = 0.62
θ = 32°
The wooden block will start to move when
the concrete incline is raised to an angle of
about 32°.
Divide both sides by cos θ.
Ratio identity.
Evaluate inverse tangent.
19. Check It Out! Example 3
Use the equation mg sinθ = μmg cosθ to
determine the angle at which a waxed wood
block on a wood incline with μ = 0.4 begins
to slide.
Set the expression for the weight component
equal to the expression for the force of friction.
mg sinθ = μmg cosθ
sinθ = μcosθ
sinθ = 0.4 cosθ
Divide both sides by mg.
Substitute 0.4 for μ.
20. Check It Out! Example 3 Continued
tanθ = 0.4
θ = 22°
The wooden block will start to move when
the concrete incline is raised to an angle of
about 22°.
Divide both sides by cos θ.
Ratio identity.
Evaluate inverse tangent.
22. Lesson Quiz: Part II
Rewrite each expression in terms of cos θ,
and simplify.
3. sin2
θ cot2
θ secθ cosθ
4. 2 cosθ
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