Lesson 11­4 Double Angle Identities Day 2
Warm­up Problems (Review of Chapter 5)
Let sin x = 4/7 and 90o < x < 180o Find...
a) cos x
Method #1: Use Pythagorean Identity to solve for cos x.
Pythagorean Identity
Input the given sin x value.
Square the fraction & rewrite the 1 as a fraction with same denominator.
Subtract on both sides
Square root both sides of the equation.
The denominator should be the same as it was for the sin x. Use the interval given in the problem to determine whether or not the cos x should be a positive or a negative number. In this case, the angle is in quadrant II (between 90 and 180 degrees) and so the cosine is negative.
b) tan x
Rewrite tangent as sin x divided by cos x.
Substitute the given value from problem for sin x and the answer for part a for cos x.
Since both are divided by 7, they can be cancelled to simplify the fraction. You do not need to rationalize the denominator.
Lesson 11­4 Double Angle Identities Day 2
Warm­up Problems (Review of Chapter 5)
Let sin x = 4/7 and 90o < x < 180o Find...
a) cos x
Method #2: Use a right triangle and SohCahToa.
Draw a right triangle where x is one of the acute angles. Since sin x = opposite/
hypotenuse, you can label the opposite side from x as 4 and the hypotenuse of the triangle as 7.
Use the Pythagorean Theorem to determine the missing side length of the triangle.
cos x = adjacent/hypotenuse from SohCahToa.
Using the side lengths in the triangle, we can determine the ratio (fraction) that cosine is equivalent to.
Since the interval given in the problem is in quadrant II, the cos x should be negative.
b) tan x
Rewrite tangent as sin x divided by cos x.
Substitute the given value from problem for sin x and the answer for part a for cos x.
Since both are divided by 7, they can be cancelled to simplify the fraction. You do not need to rationalize the denominator.
Lesson 11­4 Double Angle Identities Day 2
Objectives
Apply the double angle identities.
Lesson 11­4 Double Angle Identities Day 2
Example 1
Let csc x = 3/2 and x is in quadrant II. Find the following values without a calculator.
Each of these problems gives you a value for a trig. ratio (usually csc or sec) and another piece of information that allows you to determine what quadrant the angle is located in.
a) sin x
b) cos x
The csc x is equal to the reciprocal of the sin x. This means you can take the reciprocal of the 3/2 in the problem to get the sin x.
METHOD #1
Don't forget the problem states we are in quadrant II which means the cos x should be negative.
METHOD #2
Lesson 11­4 Double Angle Identities Day 2
c) tan x
The only part of this lesson that is new is parts d and e of this problem. The rest is a review from lesson 5­7 during first semester.
d) sin 2x
Use the double angle identity for sine.
Substitute the values from part a and b.
Multiply the fractions together and simplify the fraction.
Lesson 11­4 Double Angle Identities Day 2
Method #1
Use double angle identity with only sine function.
e) cos 2x
Substitute 2/3 from part a into problem.
Square the 2/3.
Double the 4/9.
Turn 1 into 9/9 to have a common denominator.
Subtract to fully simplify.
Method #2
Use double angle identity with only cosine function.
e) cos 2x
Substitute the cos x from part b.
Square the cos x.
Double the 5/9.
Rewrite 1 as 9/9 to get a common denominator.
Subtract to fully simplify.
Method #3
e) cos 2x
Use double angle identity with both sine and cosine function.
Substitute sin x from part a and cos x from part b.
Square both fractions.
Subtract to fully simplify.
Lesson 11­4 Double Angle Identities Day 2
Homework
Lesson 11­4 Day 2
Worksheet #1­4