Question

5. If x cos A = 8 and 15cosec A = 8 sec A,
then the value of x is​

Answer 1

Answer:

Solution:

Given,

x = cos 10° cos 20° cos 40°

x = cos 10° cos(2 × 10°) cos(22 × 10°)

We know that cos A cos 2A cos 22A…cos 2n-1A = (sin 2nA)/(2n sin A)

x = [sin 23(10°)]/ [23 sin 10°]

x = sin 80°/(8 sin 10°)

x = sin(90° – 10°)/ (8 sin 10°)

x = (1/8) [cos 10°/sin 10°)

x = (1/8) cot 10°

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Answer 2

Given:

$\[x\cos A = 8\]$ and $\[15\cos ecA = 8\sec A\]$

Solution:

Rearrange equation $\[15\cos ecA = 8\sec A\]$,

$\[ \Rightarrow \frac{{\cos ecA}}{{\sec A}} = \frac{8}{{15}}\]$

$\[ \Rightarrow \cot A = \frac{8}{{15}}\]$

$\[ \Rightarrow \tan A = \frac{{15}}{8}\]$

Know that, $\[\tan A = \frac{P}{B}\]$

Therefore. $\[\frac{P}{B} = \frac{{15}}{8}\]$

Apply Pythagoras theorem to get hypotenuse (H),

$\[\begin{array}{l}H = \sqrt {{{15}^2} + {8^2}} \\ \Rightarrow H = \sqrt {225 + 64} \\ \Rightarrow H = \sqrt {289} \\ \Rightarrow H = 17\end{array}\]$

Now calculate the value of $\[\cos x\]$,

$\[\cos A = \frac{B}{H}\]$

$\[\cos A = \frac{8}{{17}}\]$

Put, $\[\cos A = \frac{8}{{17}}\]$ in $\[x\cos A = 8\]$ and evaluate $\[x\]$,

$\[x \times \frac{8}{{17}} = 8\]$

$\[ \Rightarrow x = 17\]$

Hence, the value of $x$ is 17.