Question

Find the value of x .
cos x = 2 sin 45° cos 45° – sin 30°​

Answer 1

Step-by-step explanation:

We Have :-

$\cos(x) = 2 \sin(45) \cos(45) - \sin(30)$

To Find :-

$x$

Solution :-

$we \: know \: that \\ \\ \\ \sin(45) = \dfrac{1}{ \sqrt{2} } \\ \\ \\ \cos(45) = \dfrac{1}{ \sqrt{2} } \\ \\ \\ \cos(60) = \dfrac{1}{2} \\ \\ \\ \sin(90) = 1 \\ \\ \\ so \: now \: coming \: to \: the \: question \\ \\ \\ \cos(x) = 2 \sin(45) \cos(45) - \sin(30) \\ \\ \\ there \: are \: 2 \: ways \: to \: do \: it \\ \\ \\ 1 \: method \\ \\ \\ \cos(x) = 2 \sin(45) \cos(45) - \sin(30) \\ \\ \\ putting \: the \: values \\ \\ \\ \cos(x) = 2 \times \dfrac{1}{ \sqrt{2} } \times \dfrac{1}{ \sqrt{2} } - \dfrac{1}{2} \\ \\ \\ \cos(x) = \dfrac{2}{2} - \dfrac{1}{2} \\ \\ \\ \cos(x) = 1 - \dfrac{1}{2} \\ \\ \\ \cos(x) = \dfrac{2 - 1}{2} \\ \\ \\ \cos(x) = \frac{1}{2} \\ \\ \\ x = 60 \\ \\ \\ 2 \: method \\ \\ \\ using \: the \: identity \\ \\ \\ 2 \sin(x) \cos(x) = \sin(2x) \\ \\ \\ we \: get \\ \\ \\ \cos(x) = \sin(2 \times 45) - \sin(30) \\ \\ \\ \cos(x) = \sin(90) - \sin(30) \\ \\ \\ \cos(x) = 1 - \dfrac{1}{2} \\ \\ \\ \cos(x) = \dfrac{1}{2} \\ \\ \\ x = 60$

Answer 2

Given:-

Sin45°=$\frac{1}{\sqrt{2}}$

Cos45°=$\frac{1}{\sqrt{2}}$

Sin30°=$\frac{1}{2}$

To Find:-

The Value of x?

$\rule{200}{3}$

Proof:-

Cos x=2sin45°×cos45°-sin30°

★Putting in The Values★

→Cos x=2×$\frac{1}{\sqrt{2}}$×$\frac{1}{\sqrt{2}}$-$\frac{1}{2}$

→Cos x=2×$\frac{1×1}{\sqrt{2}× \sqrt{2}}$-$\frac{1}{2}$

→Cos x=$\frac{2}{2}$-$\frac{1}{2}$

→Cos x=$\frac{2-1}{2}$

→Cos x=$\frac{1}{2}$

[°•° Cos 60°=$\frac{1}{2}$ ]

→Cos x=Cos 60

→x=60

$\rule{200}{7}$