IXL - Solve trigonometric equations using sum and difference identities (Precalculus practice)

Solve the equation

cos

–

with

≤

First, choose the expression that is equivalent to

cos

–

sin

cos

(

)

–

cos

sin

(

)

cos

(

)

sin

(

)

sin

cos

(

)

–

cos

sin

(

)

cos

(

)

sin

(

)

Now, find all solutions to

cos

–

with

≤

Enter one exact, simplified solution in each box. You can add or remove boxes.

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Learn with an example

question

Solve the equation

cos

–

with

≤

First, choose the expression that is equivalent to

cos

–

cos

(

)

sin

(

)

cos

(

)

cos

–

sin

(

)

sin

(

)

cos

–

cos

(

)

sin

cos

(

)

cos

sin

(

)

sin

Now, find all solutions to

cos

–

with

≤

Enter one exact, simplified solution in each box. You can add or remove boxes.

key idea

For angles

and

these are the sum identities for sine, cosine, and tangent.

sin

(

)

sin

(

)

cos

(

)

cos

(

)

sin

(

)

cos

(

)

cos

(

)

cos

(

)

–

sin

(

)

sin

(

)

tan

(

)

tan

(

)

tan

(

)

–

tan

(

)

tan

(

)

For angles

and

these are the difference identities for sine, cosine, and tangent.

sin

(

–

)

sin

(

)

cos

(

)

–

cos

(

)

sin

(

)

cos

(

–

)

cos

(

)

cos

(

)

sin

(

)

sin

(

)

tan

(

–

)

tan

(

)

–

tan

(

)

tan

(

)

tan

(

)

You can use the unit circle to find the values of the trigonometric functions at key angles.

solution

You want to solve the equation

cos

–

with

≤

First, use the difference identity for cosine to find an equivalent expression for

cos

–

cos

–

cos

(

)

cos

sin

(

)

sin

Substitute this expression into the original equation.

cos

–

cos

(

)

cos

sin

(

)

sin

–

Now, solve the equation.

cos

(

)

cos

sin

(

)

sin

–

cos

(

)

sin

(

)

Substitute

cos

(

)

and

sin

(

)

–

Substitute

cos

and

sin

(

)

Simplify

–

Simplify

For angles in the range

≤

sin

(

)

–

has one solution:

So, the solution to the original equation,

cos

–

Looking for more? Try this:

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Work it out

Not feeling ready yet? These can help:

T.5Trigonometric sum and difference identities

R.11Solve trigonometric equations

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