To find
sin(2x), cos(2x), and tan(2x)
when
tan(x) = 43 and
x is in
quadrant II, we can follow these steps:
1. Understanding Quadrant II Rules
In quadrant II, the sine value is positive while the cosine value is negative.
2. Finding Sin(x) and Cos(x)
Given
tan(x) = 43, we know that
tan(x) = sin(x) / cos(x)
Letting
sin(x) = 43y and
cos(x) = -y (as cosine is negative in quadrant II), we can say:
43y / -y = 43
Using the Pythagorean theorem:
sin²(x) + cos²(x) = 1
Substituting the values:
(43y)² + (-y)² = 1
1849y² + y² = 1
1850y² = 1
y² = 1/1850
y = 1/sqrt(1850)
3. Calculating Sin(x) and Cos(x)
Now we can find:
sin(x) = 43(1/sqrt(1850)) and cos(x) = -1/sqrt(1850)
This simplifies to:
sin(x) = 43/sqrt(1850) and cos(x) = -1/sqrt(1850)
4. Finding Sin(2x) and Cos(2x)
We can use the double angle formulas:
sin(2x) = 2sin(x)cos(x)
cos(2x) = cos²(x) - sin²(x)
Substituting the values:
sin(2x) = 2(43/sqrt(1850))(-1/sqrt(1850)) = -86/1850
cos(2x) = (-1/sqrt(1850))² - (43/sqrt(1850))² = 1/1850 - 1849/1850 = -1848/1850
5. Finding Tan(2x)
Using the formula:
tan(2x) = sin(2x) / cos(2x)
So:
tan(2x) = (-86/1850) / (-1848/1850) = 86/1848
Finally, reducing
tan(2x) = 43/924
Conclusion
In summary, if
tan(x) = 43 in
quadrant II, we get:
- sin(2x) = -86/1850
- cos(2x) = -1848/1850
- tan(2x) = 43/924
This approach effectively uses trigonometric identities and knowledge of the unit circle to achieve the required calculations.