To find sin x², cos x², and tan x² given that tan x = 43 and x is in quadrant II, we start with the definition of the tangent function:
Since tan x = opposite/adjacent, we can consider a right triangle where the opposite side is 43 and the adjacent side is 1 (because tan x is positive in quadrant II and we reflect on the triangle position). Therefore, we can find the hypotenuse using the Pythagorean theorem:
Hypotenuse = √(43² + 1²) = √(1849 + 1) = √(1850)
With the sides of the triangle established, we can find the sine and cosine:
sin x = opposite/hypotenuse = 43/√1850
cos x = adjacent/hypotenuse = 1/√1850
Next, we need to determine sin x² and cos x²:
- sin x² = (sin x)² = (43/√1850)² = 1849/1850
- cos x² = (cos x)² = (1/√1850)² = 1/1850
Now, we can find tan x² using the identity tan x² = sin x²/cos x²:
tan x² = (1849/1850) / (1/1850) = 1849
In summary, we have:
- sin x² = 1849/1850
- cos x² = 1/1850
- tan x² = 1849