NIMCET 2017 — Mathematics PYQ

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Question

NIMCET 2017 — Mathematics PYQ

NIMCET | Mathematics | 2017

If $cos θ =$ $\frac{4}{5}$ and cos φ = $\frac{12}{13}$ , with $θ$ and $φ$ both in the fourth quadrant, the value of $cos(θ + φ)$ is ?

Choose the correct answer:

Explanation

Concept:
cos(A + B) = cos A cos B - sin A sin B
sin²x + cos²x = 1

In the fourth quadrant, the values for cos are positive only.

$\underline{Calculation :}$

$_{12} es i ooo$

Here, $cos θ = \frac{4}{5}$ and $cos ϕ = \frac{12}{13}$

$sin^{2} θ = 1 - cos^{2} θ$

$= 1 - (\frac{4}{5})^{2}$

$= \frac{9}{25}$

$sin θ = \pm$ 3/5

=-3/5 $..... (∵ θ$ is in the fourth quadrant)

$sin^{2} ϕ = 1 - cos^{2} ϕ$

$= 1 - (\frac{12}{13})^{2}$

$= \frac{25}{169}$
sin $ϕ = \pm$ 5/13
=-5/13

Now, cos $(θ + ϕ) = cos θ cos ϕ - sin θ sin ϕ$
$= \frac{4}{5} \times$ $\frac{12}{13} \times - (\frac{- 3}{5} \times$ $\frac{- 5}{13})$

$= \frac{33}{65}$