Question

Given sin∅ = 1/3 and 0 < ∅ < π/2 ; find tan2∅

a. (4√2)/9
b. 9/7
c. (4√2)/7
d.7/9

Answer 1

From the fundamental law of trigonometric, we have

`\sin^2(\theta) + \cos^2(\theta) = 1 \iff \cos(\theta) = \pm√(1-\sin^2(\theta))`

Since
`0 < \theta < (\pi)/(2)`, the cosine must be positive, so we choose the positive solution:

`\cos(\theta) = \sqrt{1-(1)/(9)} = (√(8))/(3)`

Now, the rule for the double angle of the tangent states that

`\tan(2\theta) = (2\sin(\theta)\cos(\theta))/(\cos^2(\theta) - \sin^2(\theta)) = (2\cdot(1)/(3)\cdot(√(8))/(3))/((8)/(9)-(1)/(9)) = ((2√(8))/(9))/((7)/(9)) = (2√(8))/(9)\cdot (9)/(7) = (2√(8))/(7)`

Answer 2

:

c. 4√2 / 7 .

Explanation:

Answer

sin Ф = 1/3.

cos Ф = √(1 - (1/3)^2) Note this will be the positive square root as

Ф is in the first quadrant.

sin 2Ф = 2 sin Ф cos Ф

= 2* 1/3 * √(1 - (1/3)^2)

= 2/3 * 2√2/3

= 4√2 / 9

cos 2Ф = 1 -2 sin^2 Ф = 1 - 2* (1/3)^2 = 7/9

tan 2 Ф = sin 2Ф / cos 2Ф = 4√2/ 9 / 7/9

= 4√2 / 9 * 9 / 7

= 4√2 / 7 (answer).