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8.4 - 8.5 Using Trig Formulas. In these sections, we will study the following topics: Using the sum and difference formulas to evaluate trigonometric functions Using the double-angle formula to evaluate trigonometric functions.

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8 4 8 5 using trig formulas
8.4 - 8.5 Using Trig Formulas

In these sections, we will study the following topics:

Using the sum and difference formulas to evaluate trigonometric functions

Using the double-angle formula to evaluate trigonometric functions


8 4 8 5 using trig formulas

The trig formulas that we will study in sections 8.4 and 8.5 are frequently used in calculus to rewrite trigonometric expressions in a form that helps you to simplify expressions and solve equations.

In this course, we will not study the derivations of these formulas, but rather we will use them to evaluate specific trig functions and simplify trig expressions.


8 4 8 5 using trig formulas

PLEASE NOTE THAT are frequently used in calculus to rewrite trigonometric expressions in a form that helps you to simplify expressions and solve equations.

The same is true for the other sum and difference formulas!


8 4 8 5 using trig formulas

Example Using Sine Sum/Difference Formula are frequently used in calculus to rewrite trigonometric expressions in a form that helps you to simplify expressions and solve equations.

Evaluate sin 75° without using a calculator. Give exact answer.

Solution

Since we are asked to evaluate the function without the calculator, we should be able to use our special reference angles and/or the quadrantal angles.

So, we want to find two special angles whose sum or difference is 75°.

Let’s use the fact that 75° = ______ + ______


8 4 8 5 using trig formulas

Example Using Sine Sum/Difference Formula are frequently used in calculus to rewrite trigonometric expressions in a form that helps you to simplify expressions and solve equations.

continued


8 4 8 5 using trig formulas

Example Using Cosine Sum/Difference Formula are frequently used in calculus to rewrite trigonometric expressions in a form that helps you to simplify expressions and solve equations.

Use a difference formula to find the exact value of cos 165°


8 4 8 5 using trig formulas

Example Using Tangent Sum/Difference Formula are frequently used in calculus to rewrite trigonometric expressions in a form that helps you to simplify expressions and solve equations.

Find the exact value of


8 4 8 5 using trig formulas

Example Using Tangent Sum/Difference Formula are frequently used in calculus to rewrite trigonometric expressions in a form that helps you to simplify expressions and solve equations.

Find the exact value of

Solution

You should recognize the form of this expression as that of the tangent sum formula, where = _________ and = _____________.

Writing this as the tangent of a sum of angles, we get:


8 4 8 5 using trig formulas

Example Using Sine Sum/Difference Formula are frequently used in calculus to rewrite trigonometric expressions in a form that helps you to simplify expressions and solve equations.

Write the expression as the sine of a single angle:


8 4 8 5 using trig formulas

Example Using Formulas to Verify Identity are frequently used in calculus to rewrite trigonometric expressions in a form that helps you to simplify expressions and solve equations.

Verify the identity:


8 4 8 5 using trig formulas

Example Using Formulas to Verify Identity are frequently used in calculus to rewrite trigonometric expressions in a form that helps you to simplify expressions and solve equations.

Verify the identity:


8 4 8 5 using trig formulas

Use whichever form is most convenient are frequently used in calculus to rewrite trigonometric expressions in a form that helps you to simplify expressions and solve equations.


8 4 8 5 using trig formulas

Solution are frequently used in calculus to rewrite trigonometric expressions in a form that helps you to simplify expressions and solve equations.

First, since  lies in Quad II, we know that:

sin  is positive

cos  is negative

tan  is negative

SA

T C

Example Using Double-Angle Formula


8 4 8 5 using trig formulas

25 are frequently used in calculus to rewrite trigonometric expressions in a form that helps you to simplify expressions and solve equations.

24

Solution (cont.)

Secondly, we can use Pythagorean Theorem to find the missing length.

Notice, we do not know the value of , nor is it necessary to find this out.


8 4 8 5 using trig formulas

Solution (cont.) are frequently used in calculus to rewrite trigonometric expressions in a form that helps you to simplify expressions and solve equations.

Finally, we will use the double angle formulas to write the three expressions. (Be careful with the signs)


8 4 8 5 using trig formulas

Example Using Double-Angle Formula are frequently used in calculus to rewrite trigonometric expressions in a form that helps you to simplify expressions and solve equations.


8 4 8 5 using trig formulas

  • End of Sections 8.4 - 8.5 are frequently used in calculus to rewrite trigonometric expressions in a form that helps you to simplify expressions and solve equations.