65.2k views
5 votes
Given tan(x)=4/3 and sin(x) is positive, find the sin(2x)

1 Answer

3 votes

First, we must acknowledge that tan(x) = sin(x)/cos(x). So, if tan(x) = 4/3, that means sin(x) and cos(x) are fractions that can be simplified into 4/3 (although we don't know just what these fractions are yet).

Because we're given that sin(x) is positive, and we know that tan(x) is positive too, we can deduce that x is in the first quadrant where both sin(x) and cos(x) are positive.

To find cos(x), we will use the Pythagorean trigonometric identity: sin²(x) + cos²(x) = 1.

To express sin(x) in terms of tan(x), we can rearrange the above relationship:

sin(x) = tan(x)/ sqrt(1 + tan²(x))

We have the value of tan(x), so we can use that to find sin(x):

sin(x) = (4/3)/ sqrt(1 + (4/3)²)

Solving this will yield:

sin(x) = 4/5

Substituting sin(x) = 4/5 into the Pythagorean trigonometric identity, we can solve for cos(x):

Cos(x) = sqrt[1 - sin²(x)]

Using sin(x) = 4/5, we find:

Cos(x) = sqrt[1 - (4/5)²] = 3/5.

Now, we can apply the double angle formula for sine that states sin(2x) = 2sin(x)cos(x)

Substituting the above sin(x) and cos(x) in, we get:

sin(2x) = 2 * (4/5) * (3/5) = 48/25

User Neal Maloney
by
8.1k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories