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# find the exact values of the expressions without solving for x.

If sin(x) = 1/4 and x is in quadrantI,find the exact values of the expressions without solving for x.

(a) sin(2x)
(b) cos(2x)
(c) tan(2x)

### Comments

cos 2x = 2cos2x - 1
sin 2x = 2sinxcosx
tan2x = (2tanx)/(1-tan2x)

### 2 Answers by Expert Tutors

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Victoria V. | Math Teacher: 18+ Years teaching/tutoring Calc, Algebra 2, Trig, GeomMath Teacher: 18+ Years teaching/tutorin...
4.9 4.9 (80 lesson ratings) (80)
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Hi again, Lauren.

A triangle is worth a zillion words...

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/      |     Use your imagination...  :-)

Remember that sin(x) = opposite side/hypotenuse.  So on our triangle, we are going to put the angle x in the bottom left vertex.  That makes the vertical leg the opposite leg = 1 and the hypotenuse = 4

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4  /  |
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Now use the Pythag Thm to determine that the horizontal leg √15.  So the cos(x)=(√15)/4

If you are not to solve for "x", can you use the trig double angle formulas?

a) sin(2x) = 2sin(x)cos(x) = 2(1/4)(√15)/4 = (1/2)(√15)/4 = (√15)/8

b) cos(2x) = cos2(x) - sin2(x) = [(√15/4)(√15/4)] - (1/4)(1/4) = 15/16 - 1/16 = 14/16 = 7/8

c) tan(2x) = [sin(2x)/cos(2x)] = [√15/8]/[7/8] = (√15)/7

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Carol H. | Experienced Mathematics Tutor w/ Master's Degree in MathExperienced Mathematics Tutor w/ Master'...
4.9 4.9 (277 lesson ratings) (277)
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Use the Pythagorean Theorem to find the length of the adjacent side in the right triangle in Quad I.
a2 + 12 = 42
a2 = 15, a = √15
cos x = (√15)/4    tan x = 1/√15 = √15/15
Now use these formulas: