Trigonometric ratios of 360 minus theta
(360 - 
θ)

In trigonometric ratios of 360 minus theta (360 - θ), the terminal sides of co-terminal angles coincide, hence their trigonometrical ratios are same.
sin(- θ ) = - sin θ
cos (- θ ) = cos θ
tan(- θ ) = - tan θ
csc(- θ ) = - csc θ
sec (- θ ) = sec θ
cot (- θ ) = - cot θ

Clearly, 360 - θ and - θ are co-terminal angles.
sin(360 - θ) = Sin(- θ)
But we know that sin(- θ) = - sinθ
$\underline{sin(360 - θ)= -sin θ} $

Again,
cos(360 - θ) = cos(- θ)
But we know that cos(- θ) =cosθ
$\underline{cos(360 - θ)= cos θ} $

Now,
tan(360 - θ) = tan(- θ)
But we know that tan(- θ) = - tanθ
$\underline{tan(360 - θ)= - tan θ} $

As we know that, csc θ = $\frac{1}{sinθ}$
∴ csc (360-θ) = $\frac{1}{sin(360-θ)}$
But sin(360-θ) = - sin(θ)
∴ csc (360-θ)= $\frac{1}{-sin θ}$
$\underline{csc(360 - θ)= -csc θ} $

And,
sec θ = $\frac{1}{cosθ}$
∴ sec (360-θ) = $\frac{1}{cos(360-θ)}$
But cos(360-θ) = cos(θ)
∴ sec (360-θ)= $\frac{1}{cos θ}$
$\underline{sec(360 - θ)= cos θ} $

Again,
cot θ = $\frac{1}{tanθ}$
∴ cot (360-θ) = $\frac{1}{tan(360-θ)}$
But tan(360-θ) = - tan(θ)
∴ cot (360-θ)= $\frac{1}{-tan θ}$
$\underline{cot(360 - θ)= -cot θ} $

Examples on Trigonometric ratios of 360 minus theta
(360 - θ)

1) Find the values of the following trigonometric ratios :
(i) sin $(315)^{0}$
Solution : sin $(315)^{0}$
315 = 360 - 45
∴ sin $(315)^{0}$ = sin (360 -45) = -sin 45
But sin 45 = $\frac{1}{\sqrt{2}}$
⇒ sin (315) = -$\frac{1}{\sqrt{2}}$

(ii) cos $(270)^{0}$
Solution : cos $(270)^{0}$
270 = 360 - 90
∴ cos $(270)^{0}$ = cos (360 -90) = cos 90
But cos(90) = 0
⇒ cos (270) = 0

(iii) tan $(300)^{0}$
Solution : tan $(300)^{0}$
300 = 360 - 60
∴ tan $(300)^{0}$ = tan (360 -60) = -tan 60
But tan(60) = $\sqrt{3}$
⇒ tan (270) = - $\sqrt{3}$



11th grade math

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