Question

Cos30°/Sec30°+Cosec30°=???​

Answer 1

Trigonometry

To solve these types of questions, use the concept listed below:

• Trigonometric ratios: They are sin, cos, tan, cot, sec, cosec.
• The standard angles of these trigonometric ratios are 0°, 30°, 45°, 60° and 90°.

Analyse the values of important angles for all the six trigonometric ratios shown in the table given below:

$\boxed{\begin{array}{c|c|c|c|c|c} \bf \theta & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin(\theta) & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos(\theta) & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan(\theta) & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec(\theta) & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec(\theta) & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot(\theta) & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0\end{array}}$

Let's head to the Question now:

$\implies \dfrac{\cos({30}^{ \circ})}{\sec({30}^{ \circ})} + \rm cosec({30}^{ \circ})$

$\implies \dfrac{ \frac{ \sqrt{3} }{2} }{ \frac{2}{ \sqrt{3} } } + 2$

$\implies \dfrac{ \sqrt{3}}{2} \times \dfrac{ \sqrt{3} }{2} + 2$

$\implies \dfrac{ \sqrt{3} \times \sqrt{3} }{2 \times 2} + 2$

$\implies \dfrac{3}{4} + 2$

$\implies \dfrac{3 + 8}{4}$

$\implies \boxed{\dfrac{11}{4}}$

Hence,

$\implies \boxed{\dfrac{\cos({30}^{ \circ})}{\sec({30}^{ \circ})} + \rm cosec({30}^{ \circ}) = \dfrac{11}{4} }$

Hence, this is our required solution.

Answer 2

Step-by-step explanation:

$\begin{gathered}\boxed{\begin{array}{c|c|c|c|c|c} \bf \theta & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin(\theta) & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos(\theta) & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan(\theta) & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec(\theta) & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec(\theta) & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot(\theta) & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0\end{array}}\end{gathered}$

$\implies \dfrac{\cos({30}^{ \circ})}{\sec({30}^{ \circ})} + \rm cosec({30}^{ \circ})⟹ sec(30$

$\implies \dfrac{ \frac{ \sqrt{3} }{2} }{ \frac{2}{ \sqrt{3} } } + 2⟹ 3$

$\implies \dfrac{ \sqrt{3}}{2} \times \dfrac{ \sqrt{3} }{2} + 2⟹$

$\implies \dfrac{ \sqrt{3} \times \sqrt{3} }{2 \times 2} + 2⟹$

$\implies \dfrac{3}{4} + 2$⟹

$\implies \dfrac{3 + 8}{4}$⟹

$\implies \boxed{\dfrac{11}{4}}$⟹

$\fbox{Hence,}$

$\implies \boxed\tt\red{{\dfrac{\cos({30}^{ \circ})}{\sec({30}^{ \circ})} + \rm cosec({30}^{ \circ}) = \dfrac{11}{4} }}$⟹

Hence, this is our required solution.