Question

How many of following are matched correctly

$\begin{array}{cc}\text{Column A}& \text{Column B}\\ \text{1. Sin}{780}^{\circ }& \text{a.}\frac{\sqrt{3}}{2}\\ \text{2. tan}{1485}^{\circ }& \text{b.}-1\\ \text{3. Cos}{1350}^{\circ }& \text{c.}-1\\ \text{4. Sec}(-{1920}^{\circ })& \text{d.}-2\\ \text{5. tan}{2340}^{\circ }& \text{e.}0\\ \text{6. cosec}(-{765}^{\circ })& \text{f.}-2\\ \text{7. Cot}{600}^{\circ }& \text{g.}\sqrt{3}\\ \text{8. Sin}{3105}^{\circ }& \text{h.}-\frac{1}{\sqrt{2}}\\ \text{9. Sin}(-{3105}^{\circ })& \text{i.}\frac{1}{\sqrt{2}}\\ \text{10. Cos}{480}^{\circ }& \text{j.}\frac{-1}{\sqrt{2}}\end{array}$
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Answer 1

Let's find the values of each trigonometric function.

(1).Sin ${780}^{\circ }$

If we add 360 to an angle, it gives the same effective angle or by rotating the radius by 360, we reach the same position.

${780}^{\circ }$ can be written as (2 $\times$ ${360}^{\circ }$ + ${60}^{\circ }$)

Sin (2 $\times$ ${360}^{\circ }$ + ${60}^{\circ }$) = sin ${60}^{\circ }$ = $\frac{\sqrt{3}}{2}$ (correctly matched)

2.tan${1485}^{\circ }$ = tan (4 $\times$ ${360}^{\circ }$ + ${45}^{\circ }$)

= tan ${45}^{\circ }$ = 1 (Not correctly matched)

3.Cos ${1350}^{\circ }$ = cos (3 $\times$ 360 + ${270}^{\circ }$)

= cos ${270}^{\circ }$ = cos (360 - ${90}^{\circ }$)

= cos ${90}^{\circ }$ = 0 (Not correctly matched)

4.Sec (-${1920}^{\circ }$) = sec ${1920}^{\circ }$ {sec (-$\theta$) = sec $\theta$}

Sec (5 $\times$ ${360}^{\circ }$ + ${120}^{\circ }$) = sec ${120}^{\circ }$

= sec (180-${60}^{\circ }$) = - sec ${60}^{\circ }$ = -2 (correctly matched)

5.tan ${2340}^{\circ }$ = tan (6 $\times$ ${360}^{\circ }$ + ${180}^{\circ }$) = tan ${180}^{\circ }$ = 0 (correctly matched)

6. cosec(-${765}^{\circ }$) = - cosec ${765}^{\circ }$ = -cosec(2 $\times$ ${360}^{\circ }$ + ${45}^{\circ }$)

= -cosec ${45}^{\circ }$ = $-\sqrt{2}$(Not correctly matched)

7. Cot ${600}^{\circ }$ = Cot(360 + 240) = cot${240}^{\circ }$

= cot(${180}^{\circ }$ + ${60}^{\circ }$) = cot ${60}^{\circ }$ = $\frac{1}{\sqrt{3}}$ (Not correctly matched)

8. Sin ${3105}^{\circ }$ = sin (8 $\times$ ${360}^{\circ }$ + ${225}^{\circ }$) = sin(${225}^{\circ }$)

= sin (${180}^{\circ }$ + ${45}^{\circ }$) = -sin ${45}^{\circ }$ = $-\frac{1}{\sqrt{2}}$ (Correctly matched)

9. Sin(-${3105}^{\circ }$) = -sin ${3105}^{\circ }$ = $-\left(\frac{-1}{\sqrt{2}}\right)$ = $\frac{1}{\sqrt{2}}$(Correctly matched)

10. Cos ${480}^{\circ }$ = cos(${360}^{\circ }$ + ${120}^{\circ }$) = cos ${120}^{\circ }$

= cos(${180}^{\circ }$ - ${60}^{\circ }$) = -cos${60}^{\circ }$ = $\frac{-1}{2}$ (Not Correctly matched)

So, 5 values are correctly matched.