Precalculus question(?)

Question

If sin(x)=7/9 and x is in quadrant I, find the exact value of sin(2x), cos(2x), and tan(2x) algebraically without solving for x.

I dont even know where to start for this one.

Answer 1

Imagine angle “x” drawn in the unit-circle.

Let point “P” be the intercept of the terminal side of the angle and the circle.

Let “C” be the center of the circle.

Let “A” be the x-coordinate of the point “P”.

Let “B” be the y-coordinate of the point “P”.

In the right triangle CPA:

PA = 7 (leg)

CP = 9 (hypotenuse)

CA = √(81 - 49) = √32 ← Pythagorean theorem

sin(x) = 7/9

cos(x) = √32/9

tan(x) = 7/√32

“2x” is called the double-angle.

To find trigonometric functions for angle “2x”, we will use double-angle identity formulas.

sin(2x) = 2•sin(x)•cos(x)

cos(2x) = cos²(x) - sin(x)

tan(2x) = 2•tan(x)/(1 - tan²(x))

Answer 2

Draw and label a diagram!

sin x = 7 / 9

cos x = √31 / 9

tan x = 9√31 / 31

All of these using Pythagoras - learn it!

Now substitute values in appropriate formula