5.4 – Sum and Difference Formulas
Learning Target: I can use sum and difference formulas to evaluate trigonometric
functions, verify identities, and solve trigonometric equations.
Essential Question: How do you simplify expressions and solve equations that
contain sums or differences of angles?
Sum and Difference Formulas:
Sin(u+v)=
Sin(u-v)=
Cos(u+v)=
Cos(u-v)=
tan(𝑢 + 𝑣) =
tan(𝑢 − 𝑣) =
EX: Find the exact value of sin 75°.
EX: Find the exact value of tan 75°
Avoiding Common Errors: When working with sum and difference formulas,
students need to be careful with radical operations. Although, a calculator gives
an approximate decimal answer, the exact value requires the radical form.
EX: Find the exact value of cos
𝜋
12
EX: Find the exact value of cos 25° cos 20° − sin 25° sin 20°
EX: Write sin(arctan 1 + arccos x) as an algebraic expression.
We need to recognize that this is the _____ formula for ______. sin(𝑢 + 𝑣) =
__________________________, where u=____________ and v=____________.
We need to create two triangles that represent the situation.
(draw two triangles on the whiteboard)
𝜋
EX: Simplify the cofunction identity sin (𝑥 − )
2
3𝜋
EX: Simplify the expression sin (
2
− 𝜃)
𝜋
EX: Simplify the expression tan (𝜃 − )
4
𝜋
EX: Find all of the solutions of sin (𝑥 + ) + sin (𝑥 −
2
3𝜋
2
) = 1 in the interval
[0,2𝜋].
We should recognize that this is an example of the ___ identity for ______.
Remember, we can always check our solutions on the ______________________
HW Pg.404 3-69 3rds, 97-104