vocabulary reduction identity. key concept 1 example 1 evaluate a trigonometric expression a. find...
TRANSCRIPT
• reduction identity
Evaluate a Trigonometric Expression
A. Find the exact value of cos 75°.
30° + 45° = 75°
Cosine Sum Identity
Evaluate a Trigonometric Expression
Multiply.
Combine the fractions.
Answer:
Evaluate a Trigonometric Expression
B. Find the exact value of tan .
Write as the sum or difference of angle measures
with tangents that you know.
Evaluate a Trigonometric Expression
Tangent Sum Identity
Simplify.
Rationalize the denominator.
Evaluate a Trigonometric Expression
Answer:
Multiply.
Simplify.
Simplify.
Use a Sum or Difference Identity
A. ELECTRICITY An alternating current i in amperes in a certain circuit can be found after t seconds using i = 4 sin 255t, where 255 is a degree measure. Rewrite the formula in terms of the sum of two angle measures.
Rewrite the formula in terms of the sum of two angle measures.
i = 4 sin 255t Original equation
= 4 sin (210t + 45t) 255t = 210t + 45t
The formula is i = 4 sin (210t + 45t).
Answer: i = 4 sin (210t + 45t)
Use a Sum or Difference Identity
B. ELECTRICITY An alternating current i in amperes in a certain circuit can be found after t seconds using i = 4 sin 255t. Use a sum identity to find the exact current after 1 second.
Use a sum identity to find the exact current after 1 second.
i= 4 sin (210t + 45t)Rewritten equation
= 4 sin (210 + 45)t = 1
= 4[sin(210)cos(45) + cos(210)sin(45)]Sine Sum Identity
Use a Sum or Difference Identity
Simplify.
Substitute.
The exact current after 1 second is amperes.
Answer: amperes
Multiply.
Rewrite as a Single Trigonometric Expression
A. Find the exact value of
Simplify.
Tangent Difference Identity
Answer:
Substitute.
Rewrite as a Single Trigonometric Expression
Answer:
B. Simplify
Simplify.
Rewrite as fractions with a common denominator.
Sine Sum Identity
Write as an Algebraic Expression
Write as an algebraic
expression of x that does not involve
trigonometric functions.
Applying the Cosine Sum Identity, we find that
Write as an Algebraic Expression
If we let α = and β = arccos x, then sin α =
and cos β = x. Sketch one right triangle with an acute
angle α and another with an acute angle β. Label the
sides such that sin α = and cos β = x. Then use
the Pythagorean Theorem to express the length of
each third side.
Write as an Algebraic Expression
Using these triangles, we find that
= cos α or ,
cos (arccos x) = cos β or x,
= sin α or , and
sin (arccos x) = sin β or .
Write as an Algebraic Expression
Now apply substitution and simplify.
Write as an Algebraic Expression
Answer:
Verify Cofunction Identities
Verify cos (–θ) = cos θ.
cos (–θ) = cos (0 – θ) Rewrite as a difference.
= cos 0 cos θ + sin 0 sin θCosine Difference Identity
= 1 cos θ + 0 sin θcos 0 = 1 and sin 0 = 0
= cos θ Multiply.
Answer: cos (–θ) = cos (0 – θ) = cos 0 cos θ + sin 0 sin θ = 1 cos θ + 0 sin θ = cos θ
Verify Reduction Identities
Simplify.
A. Verify .
Cosine Difference Identity
Verify Reduction Identities
Answer:
B. Verify tan (x – 360°) = tan x.
Verify Reduction Identities
Tangent Difference Identity
tan 360° = 0
Simplify.
Answer:
Solve a Trigonometric Equation
Find the solutions of
on the interval [ 0, 2).
Original equation
Sine Sum Identity and Sine Difference Identity
Solve a Trigonometric Equation
Substitute.
Solve for cos x.
Simplify.
Divide each side by 2.
Solve a Trigonometric Equation
CHECK The graph of
has zeros at on the interval [ 0, 2π).
Answer:
On the interval [0, 2π), cos x = 0 when x =