Finding Other Trig Function using Basic Trig Numbers

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Question: If sin(x) = 1/5 and x is in quadrant I, find the exact values of the expressions without solving for x.
If sin(x) = 1/8 and x is in quadrant I, find the exact values of the expressions without solving for x.A) sin(2x)B) cos(2x)C) tan(2x)

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Answered By:

Shin C.
K-12 Math, Calculus, and SAT Math I and II Expert
More information: https://www.wyzant.com/Tutors/CA/Rancho_Cucamonga/9777184/?utm_source=youtube&utm_medium=organic&utm_campaign=aae_video

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Written Explanation:

Let's continue finding tan(2x).We earlier stated in the video about the formula tan(2x) = ( 2 * tan(x) ) / ( 1 - tan^2(x) ).Tangent is defined as tan(x) = opposite / adjacent, so for this problem, tan(x) = sqrt(6) / 12.Plugging in appropriate numbers, tan(2x) = ( sqrt(6) / 6) / ( 1 - 6/144 ) = 12 * sqrt(6) / 19 ....... (ANS).The exact same process can be used to find when sin(x) = 1/8. Create a triangle, and identify 1 as the vertical length, 8 as the hypotenuse, and by Pythagorean theorem, adjacent = sqrt( 8^2 - 1^2) = sqrt(63). So, sin(x) = 1/8. cos(x) = sqrt(63) / 8, and tan(x) = sqrt(63) / 63 = sqrt(7) / 21 (radicalized it).Therefore, sin(2x) = 2 * 1/8 * sqrt(63) / 8 = 3 * sqrt(7) / 32cos(2x) = 2 * ( sqrt(63) / 8 )^2 - 1 = 31/32tan(2x) = 2 * ( sqrt(7) / 21 ) / ( 1 - ( sqrt(7) / 21 )^2 ) = 3 * sqrt(7) / 31 ››››› (END)Hope this helps! Let me know in the comments of any feedback!
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