SOLUTION: I need help finding the exact values Use a double-angle identity to find the exact value of each expression. 1. sin 120° 2. tan 60° 3. cos 4π/3 4. sin

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Question 1139293: I need help finding the exact values
Use a double-angle identity to find the exact value of each expression.

1. sin 120°

2. tan 60°

3. cos 4π/3

4. sin 5π/3''.
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Use a double-angle identity to find the exact value of each expression.
1.
sin%28+120+%29..........use identity sin%28+2a%29+=+2sin%28a%29+cos%28a%29
=sin%28+2%2A60%29
=2sin%2860%29+cos%2860%29
=2sqrt%283%29%2F2%2A%281%2F2%29
=cross%282%29sqrt%283%29%2F2%2A%281%2Fcross%282%29%29
=sqrt%283%29%2F2

2.
tan+%2860%29°
=tan+%282%2A30%29+
=sin+%282%2A30%29+%2Fcos%282%2A30%29.........................sin%28+2a%29+=+2sin%28a%29+cos%28a%29 and cos+%282a%29+=+cos%5E2%28a%29+-+sin%5E2%28a%29+
=%282sin%2830%29+cos%2830%29%29%2F%28cos%5E2%2830%29+-+sin%5E2%2830%29+%29
=%282%281%2F2%29+%28sqrt%283%29%2F2%29%29%2F%283%2F4+-+1%2F4+%29
=%281%2A+%28sqrt%283%29%2F2%29%29%2F%282%2F4+%29
=4sqrt%283%29%2F4
=sqrt%283%29


3.
+cos+%284pi%2F3+%29
=cos+%282%2A%282pi%2F3+%29%29
= cos%5E2%282pi%2F3%29+-+sin%5E2%282pi%2F3%29+
=1%2F4-3%2F4
=-2%2F4
=-1%2F2


4.
sin%28+5pi%2F3+%29
=sin%28+3pi%2F3+%2B2pi%2F3%29
=sin%28+pi+%2B2pi%2F3%29..........use sin+%28a+%2B+b%29+=+sin+%28a%29+cos%28+b%29+%2B+sin+%28b%29+cos+%28a%29
=sin%28+pi%29cos%282pi%2F3%29+%2Bsin%282pi%2F3%29cos%28pi%29
=0%2A%28-1%2F2%29+%2B%28sqrt%283%29%2F2%29%28-1%29
=-sqrt%283%29%2F2



Answer by ikleyn(52805) About Me  (Show Source):
You can put this solution on YOUR website!
.

            You do not need to use "a double-angle identity to find the exact value of each expression."

            Every student,  studying trigonometry,  must know and keep in his  (or her)  memory
            the values of trigonometric functions of the given remarkable  (particular)  angles.

            In the same way as every student must know the multiplication table.



1.   sin(120°) = sqrt%283%29%2F2.

2.   tan(60°) = sqrt%283%29.

3.   cos%284pi%2F3%29 = -1%2F2.

4.   sin%285pi%2F3%29 = -sqrt%283%29%2F2.


See these sources

            https://en.wikipedia.org/wiki/Trigonometric_functions
            https://en.wikipedia.org/wiki/Trigonometric_functions

            https://en.wikipedia.org/wiki/Unit_circle
            https://en.wikipedia.org/wiki/Unit_circle

or learn it from any textbook on Trigonometry.


Happy learning this "multiplication table" of Trigonometry !