Question

Please help me out :P

Answer 1

cos θ = $\frac{-4\sqrt{65} }{65}$, sin θ = $\frac{-7\sqrt{65} }{65}$, cot  θ  = 4/7, sec  θ = $\frac{-\sqrt{65} }{4}$, cosec  θ  = $\frac{-\sqrt{65} }{7}$

What are trigonometric ratios?

Trigonometric Ratios are values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.

Sin θ: Opposite Side to θ/Hypotenuse

Tan θ: Opposite Side/Adjacent Side & Sin θ/Cos

Cos θ: Adjacent Side to θ/Hypotenuse

Sec θ: Hypotenuse/Adjacent Side & 1/cos θ

Analysis:

tan θ = opposite/adjacent = 7/4

opposite = 7, adjacent = 4.

we now look for the hypotenuse of the right angled triangle

hypotenuse = $\sqrt{7^{2} + 4^{2} }$ = $\sqrt{49+16}$ = $\sqrt{65}$

sin θ = opposite/ hyp = $\frac{7}{\sqrt{65} }$

Rationalize, $\frac{7}{\sqrt{65} }$ x $\frac{\sqrt{65} }{\sqrt{65} }$ = $\frac{7\sqrt{65} }{65}$

But θ is in the third quadrant(180 - 270) and in the third quadrant only tan and cot are positive others are negative.

Therefore, sin θ = - $\frac{7\sqrt{65} }{65}$

cos   θ  = adj/hyp = $\frac{4}{\sqrt{65} }$

By rationalizing and knowing that cos  θ  is negative, cos θ  = -$\frac{-4\sqrt{65} }{65}$

cot θ  = 1/tan θ  = 1/7/4 = 4/7

sec θ  = 1/cos θ  = 1/$\frac{4}{\sqrt{65} }$ = -$\frac{-\sqrt{65} }{4}$

cosec θ  = 1/sin θ  = 1/$\frac{\sqrt{65} }{7}$ = $\frac{-\sqrt{65} }{7}$

Learn more about trigonometric ratios: brainly.com/question/24349828

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