cos θ = [tex]\frac{-4\sqrt{65} }{65}[/tex], sin θ = [tex]\frac{-7\sqrt{65} }{65}[/tex], cot θ = 4/7, sec θ = [tex]\frac{-\sqrt{65} }{4}[/tex], cosec θ = [tex]\frac{-\sqrt{65} }{7}[/tex]
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 = [tex]\sqrt{7^{2} + 4^{2} }[/tex] = [tex]\sqrt{49+16}[/tex] = [tex]\sqrt{65}[/tex]
sin θ = opposite/ hyp = [tex]\frac{7}{\sqrt{65} }[/tex]
Rationalize, [tex]\frac{7}{\sqrt{65} }[/tex] x [tex]\frac{\sqrt{65} }{\sqrt{65} }[/tex] = [tex]\frac{7\sqrt{65} }{65}[/tex]
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 = - [tex]\frac{7\sqrt{65} }{65}[/tex]
cos θ = adj/hyp = [tex]\frac{4}{\sqrt{65} }[/tex]
By rationalizing and knowing that cos θ is negative, cos θ = -[tex]\frac{-4\sqrt{65} }{65}[/tex]
cot θ = 1/tan θ = 1/7/4 = 4/7
sec θ = 1/cos θ = 1/[tex]\frac{4}{\sqrt{65} }[/tex] = -[tex]\frac{-\sqrt{65} }{4}[/tex]
cosec θ = 1/sin θ = 1/[tex]\frac{\sqrt{65} }{7}[/tex] = [tex]\frac{-\sqrt{65} }{7}[/tex]
Learn more about trigonometric ratios: brainly.com/question/24349828
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