Find the values of trigonometric ratios

Find the values of trigonometric ratios :
1) sin (315)
Solution : sin $(315)^{0}$
sin $(315)^{0}$ = sin (90 x 3 + 45)
Since 315 lies in the 4th quadrant and in this quadrant sine function is negative , also 3 is an odd integer.
∴ sin $(315)^{0}$ = sin (90 x 3 + 45) = - cos 45 = $\frac{-1}{\sqrt{2}}$

2) cos (210)
Solution : cos $(210)^{0}$
cos $(210)^{0}$ = cos(90 x 2 + 30)
Since 210 lies in the 3rd quadrant and in this quadrant cosine function is negative.Also the multiple of 90 is even.
∴ cos $(210)^{0}$ = cos (90 x 2 + 30) = - cos 30 = $\frac{-\sqrt{3}}{2}$

3) cos (480)
Solution : cos $(480)^{0}$
cos $(480)^{0}$ = cos(90 x 5 + 30)
Since 480 lies in the 2nd quadrant and in this quadrant cosine function is negative.Also the multiple of 90 is odd.
∴ cos $(480)^{0}$ = cos (90 x 5 + 30) = -sin 30 = $\frac{-1}{2}$

4) csc (390)
Solution : csc $(390)^{0}$
csc $(390)^{0}$ = csc (90 x 4 + 30)
Since 390 lies in the 1st quadrant and in this quadrant sine function is positive so cosecant function is also positive , also 4 is an even integer.
∴ csc $(390)^{0}$ = csc (90 x 4 + 30) = csc 30 = 2

5) tan $\frac{19π}{3}$
Solution : tan $\frac{19π}{3}$
$\frac{19π}{3} = \left ( \frac{19}{3} \times 180\right ) $ = 1140
tan $(1140)^{0}$ = tan (90 x 12 + 60)
Since 1140 lies in the 1st quadrant and in this quadrant tangent function is positive , also 12 is an even integer.
∴ tan $\frac{19π}{3}$= tan (90 x 12 + 60) = tan 60 = $\sqrt{3}$

6) Prove that cos $(510)^{0}$ cos $(330)^{0}$ + sin $(390)^{0}$ cos $(120)^{0}$ = -1
Solution :
We have,
cos $(510)^{0}$ cos $(330)^{0}$ + sin $(390)^{0}$ cos $(120)^{0}$
= cos(90 x 5 + 60) cos(90 x 3 + 60) + sin(90 x 3 + 30) cos(90 x 1 + 30)
= (- sin 60) (sin 60) + (sin 30)(- sin 30)
= $\frac{-\sqrt{3}}{2} \times \frac{\sqrt{3}}{2} + \frac{1}{2} \times \frac{-1}{2}$

= $\frac{-3}{4} - \frac{1}{4}$
= -1
∴ cos $(510)^{0}$ cos $(330)^{0}$ + sin $(390)^{0}$ cos $(120)^{0}$ = -1

Practice on find the values of trigonometric ratios

Q.1 Find the values of the following trigonometric ratios.
1) cot $(570)^{0}$
2) cos $(270)^{0}$
3) sin $\frac{5π}{3}$
4) tan $\frac{11π}{6}$
5) cos $(1755)^{0}$
11th grade math

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