The tan θ and cos θ values of 17π/12 are illustrations of trigonometry ratios
The values of tan θ and cos θ are 2 + √3 and (√2 - √6)/4, respectively
We have:
θ = 17π/12
Express as sum
θ = 9π/12 + 8π/12
Simplify
θ = 3π/4 + 2π/3
The above becomes
tan(17π/12) = tan(3π/4 + 2π/3)
Using the sum formula, we have:
tan(A + B) = [tan(A) + tan(B)]/[1 - tan(A)tan(B)]
Substitute known values
tan(3π/4 + 2π/3) = [tan(3π/4) + tan(2π/3)]/[1 - tan(3π/4)tan(2π/3)]
Evaluate the expression
tan(3π/4 + 2π/3) = [-1 - √3]/[1 - (-1)(-√3)]
Evaluate the product
tan(3π/4 + 2π/3) = [-1 - √3]/[1 - √3]
Rationalize
[tex]tan(\frac{3\pi}4 + \frac{2\pi}3) = \frac{-1 - \sqrt3}{1 - \sqrt3} * \frac{1 + \sqrt3}{1 + \sqrt3}[/tex]
Evaluate the product
[tex]tan(\frac{3\pi}4 + \frac{2\pi}3) = \frac{-(1 + \sqrt3)^2}{1 - 3}[/tex]
This gives
[tex]tan(\frac{3\pi}4 + \frac{2\pi}3) = \frac{-(1 + 3 + 2\sqrt 3)}{-2}[/tex]
[tex]tan(\frac{3\pi}4 + \frac{2\pi}3) = \frac{-(4 + 2\sqrt 3)}{-2}[/tex]
Divide
[tex]tan(\frac{3\pi}4 + \frac{2\pi}3) = 2 + \sqrt 3[/tex]
So, we have:
tan(17π/12) = 2 + √3
We have:
θ = 17π/12
Express as difference
θ = 9π/4 - 5π/6
The above becomes
cos(17π/12) = cos(9π/4 - 5π/6)
Using the difference formula, we have:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
So, we have
cos(17π/12) = cos(9π/4)cos(5π/6) + sin(9π/4)sin(5π/6)
Evaluate
[tex]cos(\frac{17\pi}{12}) = \frac{\sqrt 2}{2} * - \frac{\sqrt 3}{2} + \frac{\sqrt 2}{2} * \frac 12[/tex]
Evaluate
[tex]cos(\frac{17\pi}{12}) = \frac{\sqrt 2}{2} (- \frac{\sqrt 3}{2} + \frac 12)[/tex]
Evaluate the difference
[tex]cos(\frac{17\pi}{12}) = \frac{\sqrt 2}{2} (\frac{1 - \sqrt 3}{2})[/tex]
Expand
cos(17π/12) = (√2 - √6)/4
Hence, the values of tan θ and cos θ are 2 + √3 and (√2 - √6)/4, respectively
Read more about trigonometry ratios at:
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