Find the exact value of the Double-Angle Formula tan (2A) given the cos (A) = -5/13.

Question

a. 

169/120

b. 

119/120

c. 

169/119

d. 

120/119

Answer 1

tan2A =+ or - 120/119

if cosA = -5/13,

A is in quadrant 2 or 3 where all cosines are negative

but 2A not necessarily the same, but negative in this case

sinA = -12/13 as it's a 5-12-13 right angle for the reference angle

tanA = sinA/cosA = 12/5

tan2A = 2tan(A)/(1-tan^2(A))= (24/5)/(1-144/25) = about - 1.008403361

tan2A = exactly - 120/119

go with option D,

but oddly B is closest to a negative solution, so it's reallyt the "best" answer

Answer 2

The double angle formula for tangent is:

Given that cos(A) = -5/13 and cosine is the ration of adjacent to hypotenuse, we can draw a triangle like this to represent the situation:

Using the Pythagorean Theorem, we can solve for "y" to get y = 12. Note that the angle could also possibly be in Q3 and still have cos(A) = -5/12 meaning that "y" could also be -12 as well.

Given that y = ±12 and x = -5, then tan(A) = ±12/5 (opposite/adjacent)

Using the double angle formula:

tan(2A) = 2tan(A)/(1 - (tan(A))²) and plugging in the values we got for tan(a):

Using y = 12:

tan(2A) = 2(12/-5)/(1 - (12/-5)²) = (-24/5)/(1 - 144/25) = (-24/5)/(25/25 - 144/25) = (-24/5)/(-119/25) = 120/119

or, using y = -12:

tan(2A) = 2(-12/-5)/(1 - (-12/-5)²) = (24/5)/(1 - 144/25) = (24/5)/(25/25 - 144/25) = (24/5)/(-119/25) = -120/119

So tan(2A) = ± 120/119

Answer 3

Determine sin A = ?

Determine tan A = ?

Use appropriate double angle formula.