If sin x=1/3, x in quadrant 1, then find (without finding x)

Question

Sin (2x)=?

Cos (2x)=?

Tan (2x)=?

How would I solve this equation? If possible, please explain how you solved it.

Answer 1

Recall sin(a+b) = sin(a)cos(b) + cos(a)sin(b) and cos(a+b) = cos(a)cos(a) - sin(b)sin(b):

sin(2x)

= sin(x+x)

= sin(x)cos(x) + cos(x)sin(x)

= 2sin(x)cos(x)

cos(2x)

= cos(x+x)

= cos(x)cos(x) - sin(x)sin(x)

= cos²(x) - sin²(x)

tan(2x)

= sin(2x)/cos(2x)

If sin(x)=1/3, then we can construct a triangle of height 1 which is opposite of angle x and hypotenuse 3. By using the Pythagorean Theorem, the adjacent leg is length √(3²-1²) = √(9-1) = √8 = 2√2.

Because cos(x) = adjacent/hypotenuse, then cos(x) = (2√2)/3.

We can then use our values for sin(x) and cos(x) to determine sin(2x), cos(2x), and tan(2x):

sin(2x) = 2sin(x)cos(x) = 2(1/3)((2√2)/3) = (4√2)/9

cos(2x) = cos²(x) - sin²(x) = ((2√2)/3)² - (1/3)² = 8/9 - 1/9 = 7/9

tan(2x) = sin(2x)/cos(2x) = ((4√2)/9)/(7/9) = (4√2)/7

Hope this was helpful!

Answer 2

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