Download Task: DOUBLE-ANGLE IDENTITIES FOR SINE, COSINE, AND

Georgia Department of Education
Georgia Standards of Excellence Framework Accelerated GSE Pre-Calculus • Unit 4 Mathematics Accelerated GSE Pre-Calculus Unit 4:
Trigonometric Identities Richard Woods, State School Superintendent July 2015 
All Rights Reserved
Name: ______________________________________ Date: _____________
Task: DOUBLE-ANGLE IDENTITIES FOR SINE, COSINE, AND TANGENT
Before we begin… Evaluate the following expressions without a calculator.
1a) 𝑐𝑜𝑠45° = __________
1b) 𝑐𝑜𝑠90°= __________
2a) 𝑠𝑖𝑛60° = __________
2b) 𝑠𝑖𝑛120° = _____________
𝜋
3a) 𝑠𝑖𝑛 = __________
6
𝜋
4a) 𝑐𝑜𝑠 2 = __________
5a) 𝑡𝑎𝑛
5𝜋
6
= __________
6a) 𝑡𝑎𝑛45° = __________
𝜋
3b) 𝑠𝑖𝑛 = __________
3
4b) 𝑐𝑜𝑠𝜋 = __________
5b) 𝑡𝑎𝑛
5𝜋
3
= __________
6b) 𝑡𝑎𝑛90° = __________
* Now, study the expressions in parts a and b of the problems above. They are related, but how? How do they
differ? (Hint: Look at the value in 1a) and then look at the value in 1b) ). ________________________________
_____________________________________________________________________________________________
* Based on the observations above, which of the following statements do you THINK are TRUE and which ones
do you think are FALSE?
7) If an angle is doubled (i.e. from 45° to 90°) then the sine value of the angle is doubled, too. __________
8) If an angle is halved (i.e. from 90° to 45°) then the cosine value of the angle is halved, too. __________
9) 2 ∙ sin x = sin 2x
__________
10) “Double the angle” and “double the sine value” really mean the same thing.
__________
11) The equation 2cos x = cos 2x has no solutions. (Hint: test this by substituting a value
for x)
__________
12) Doubling angles does NOT double their trig values.
__________
Georgia Department of Education
Georgia Standards of Excellence Framework Accelerated GSE Pre-Calculus • Unit 4 Mathematics Accelerated GSE Pre-Calculus Unit 4:
Trigonometric Identities Richard Woods, State School Superintendent July 2015 
All Rights Reserved
One more thing….take a look at the following graphs. Can you identify the two equations that have been
graphed? Write them on opposite sides of the “equals” sign below:
____________________ = ____________________
What are your opinions of the following questions:
13) Is the equation that you wrote on the lines above an identity? _____________________________________
14) Does the equation above have solutions? _____________________________________________________
15) Does the graph above change your opinion of any of the true or false statements from the previous page?
__________________________________________________________
 Hopefully you now know that if an angle is doubled, we know very little about what
happens to its trig values. One thing we are certain of, however, is that doubling an
angle does NOT double its trig values (most of the time!)
Georgia Department of Education
Georgia Standards of Excellence Framework Accelerated GSE Pre-Calculus • Unit 4 Mathematics Accelerated GSE Pre-Calculus Unit 4:
Trigonometric Identities Richard Woods, State School Superintendent July 2015 
All Rights Reserved
Try this:
A) Take the equation sin(𝑥 + 𝑦) = sin 𝑥 cos 𝑦 + cos 𝑥 sin 𝑦 and rewrite the identity replacing all ys with x.
_______________________________________________________________________________________
B) Now, combine like terms on the left side of the equals sign and on the right side, in order to simplify
what was written in A above.
_______________________________________________________________________________________
C) Use the new mathematical identity you found in B above to complete the following statement (Hint:
use the box of words to the right of the statement to help finish your thought):
sine
To find the sine value of an angle that I’ve doubled, I can…
_________________________________________________________________________
cosine
sum
product
double
D) Now repeat the process of A – B above for cos(x + y) in the space below.
half
identity
value
argument
Georgia Department of Education
Georgia Standards of Excellence Framework Accelerated GSE Pre-Calculus • Unit 4 Mathematics Accelerated GSE Pre-Calculus Unit 4:
Trigonometric Identities Richard Woods, State School Superintendent July 2015 
All Rights Reserved
===============================================================================
* Cosine
actually has more than one double identity formula (more than the
identity we discovered earlier)
Example: Verify this identity (i.e. show that the expression on the left side really does equal the expression on
the right side)
𝑐𝑜𝑠 2 𝑥 − 𝑠𝑖𝑛2 𝑥 = 2𝑐𝑜𝑠 2 𝑥 − 1
Example: Verify this identity (i.e. show that the expression on the left side really does equal the expression on
the right side)
𝑐𝑜𝑠 2 𝑥 − 𝑠𝑖𝑛2 𝑥 = 1 − 2𝑠𝑖𝑛2 𝑥
* Earlier, you discovered that 𝒄𝒐𝒔𝟐𝒙 = 𝒄𝒐𝒔𝟐 𝒙 − 𝒔𝒊𝒏𝟐 𝒙. Use the information that you have
discovered above to fill in the equations below (these are NEW identities)
* cos(2x) = cos 2 x − sin2 x
OR
OR *cos(2x) = _____________________
* cos(2x) = ___________________
Georgia Department of Education
Georgia Standards of Excellence Framework Accelerated GSE Pre-Calculus • Unit 4 Mathematics Accelerated GSE Pre-Calculus Unit 4:
Trigonometric Identities Richard Woods, State School Superintendent July 2015 
All Rights Reserved
====================================================================
 You have written a double-angle identity for tangent already (earlier in the task). Try
simplifying 𝑡𝑎𝑛2𝑥 =
Step 1) tan2x =
Step 2) tan2x =
Step 3) tan2x =
Step 4) tan2x =
Step 5) tan2x =
𝑠𝑖𝑛2𝑥
𝑐𝑜𝑠2𝑥
to get the same thing. Use the clues below to help you.
𝑠𝑖𝑛2𝑥
𝑐𝑜𝑠2𝑥
𝑐𝑜𝑠 2 𝑥− 𝑠𝑖𝑛2 𝑥
(divide top and bottom by 𝑐𝑜𝑠 2 𝑥)
𝑐𝑜𝑠2 𝑥− 𝑠𝑖𝑛2 𝑥
𝑐𝑜𝑠2 𝑥
𝑐𝑜𝑠2 𝑥 𝑠𝑖𝑛2 𝑥
−
𝑐𝑜𝑠2 𝑥 𝑐𝑜𝑠2 𝑥
(simplify the numerator and split the denominator)
(simplify)
Georgia Department of Education
Georgia Standards of Excellence Framework Accelerated GSE Pre-Calculus • Unit 4 Mathematics Accelerated GSE Pre-Calculus Unit 4:
Trigonometric Identities Richard Woods, State School Superintendent July 2015 
All Rights Reserved
==================================================================
DERIVING HALF-ANGLE IDENTITIES
 To derive the half angle identities, you have to use the double angle identities that you
already derived.
Step 1) Pick one of the alternate forms of the cosine double angle identities:
𝑐𝑜𝑠2𝑥 = 2𝑐𝑜𝑠 2 𝑥 − 1
Step 2) Get 𝑐𝑜𝑠 2 𝑥 by itself
Step 3) Now take the √ of both sides.
Step 4) This version is not user-friendly for half-angles. It looks like it only works for
𝑢
regular angles. Sooo….substitute everywhere there is an x in the equation. Simplify.
2
NOW repeat the steps above using another alternate cos2x identity and simplify.
(There is a tangent half-angle identity that exists but it’s not needed at this time)
Georgia Department of Education
Georgia Standards of Excellence Framework Accelerated GSE Pre-Calculus • Unit 4 Mathematics Accelerated GSE Pre-Calculus Unit 4:
Trigonometric Identities Richard Woods, State School Superintendent July 2015 
All Rights Reserved