Given: tanx = 2.4, π/2<x<3π/2; Find: sinx and cotx
sin(x) = -12/13 and cot(x) = 5/12
Ty!! :D
To find the values of sinx and cotx given tanx = 2.4, we can use the trigonometric relationships between the different trigonometric functions.
1) We know that tanx = sinx/cosx. Since tanx = 2.4, we have the equation: 2.4 = sinx/cosx.
2) We are asked to find sinx. To do this, we can solve the equation in step 1 for sinx. Multiply both sides of the equation by cosx: 2.4 * cosx = sinx.
3) Now we need to find the value of cosx. From the given range π/2<x<3π/2, we know that cosine is negative in this interval. Therefore, we can use the fact that sin^2x + cos^2x = 1 to find the value of cosx. Since tanx = 2.4, we can use the identity tan^2x + 1 = sec^2x to find cosx.
Substitute tanx = 2.4 into the equation tan^2x + 1 = sec^2x: 2.4^2 + 1 = sec^2x.
Simplify: 5.76 + 1 = sec^2x.
Combine: 6.76 = sec^2x.
Take the square root of both sides: √6.76 = secx.
4) Since cosine is negative in the given interval, the value of cosx is negative. Therefore, we have cosx = -√6.76.
5) Substitute cosx = -√6.76 back into the equation in step 2: 2.4 * (-√6.76) = sinx.
Multiply: -2.4 √6.76 = sinx.
Now we have found the values of sinx and cosx in terms of the given tanx.
Next, to find the value of cotx, we can use the reciprocal relationship between cotangent and tangent.
6) We know that cotx = 1/tanx. Since tanx = 2.4, we have cotx = 1/2.4.
Divide: cotx = 0.4167 (rounded to four decimal places).
Therefore, the approximate values of sinx, cosx, and cotx are as follows:
sinx ≈ -2.4 √6.76
cosx ≈ -√6.76
cotx ≈ 0.4167