Question

evaluate cos60 cos30 - sin60 sin30

Answer 1

Hi,



Cos60° = 1/2



Sin60° = ✓3/2



Cos30° = ✓3/2


And,


Sin30° = 1/2



Therefore,



Cos 60° × Cos30 - Sin60° × Sin30°



=> 1/2 × ✓3/2 - ✓3/2 × 1/2




=> ✓3/4 - ✓3/4



=> ✓3-✓3/4


=> 0/4



=> 0

Answer 2

On solving the equation the answer is 0.

The values of sin 30°, cos 30°, sin 60° and cos 60° can be computed by deriving the values from an equilateral triangle and by applying different trigonometric rules for the values of sin and cos.

Let us consider an equilateral triangle whose sides are equal in length and it is known that the angle formed at each corner is said to be 60°.

Thus, the angle ∠ABC is 60° .

If we drop a perpendicular from A such that the line bisects the side BC and also bisects the angle formed, that is, BAC.

Thus, the ∠BAC is 30°

Consider each of the sides to be of length 2a where a is an arbitrary constant. Then, we have:

$BD=\frac{BC}{2} =\frac{2a}{2}$

$\implies BD=a$

The sin rule can be applied to triangles where there are two given angles and one side (AAS and ASA) which states that the sides of a triangle are proportional to the sine of its opposite angles.

By applying the Pythagorean theorem the adjacent side is calculated. We have:

$AB^{2} =AC^{2} +AD^{2}$

$\implies AD^{2}= AB^{2} -AC^{2}$

By substituting the known values we get:

$\implies AD= \sqrt{2a^{2} -a^{2}}$  [∵$AB=2a$ and $AC=a$]

$\implies AD= \sqrt{3}\;a$

Now, the trigonometric rules are to be applied. We know that sine of an angle is equivalent to the ratio of the opposite side to its hypotenuse.

Similarly, cosine of angle is given by the ratio of the adjacent side to the hypotenuse.

Hence, we have:

$sin\;\theta=\frac{opposite\;side}{hypotenuse}$

We need to find sin 30° and sin 60° so by substituting the values we get:

$sin\;30=\frac{a}{2a}$

$\implies sin\;30=\frac{1}{2} \;\;-----(1)$

Similarly,

$sin\;60=\frac{\sqrt{3}a }{2a} \;\;-----(2)$

We also know that,

$cos\;\theta=\frac{adjescent\;side}{hypotenuse}$

We also need to find cos 30° and cos 60° so by substituting the values we get:

$cos\;30=\frac{\sqrt{3} a}{2a}$

$\implies cos\;30=\frac{\sqrt{3} }{2} \;\;-----(3)$

Similarly,

$cos\;60=\frac{a }{2a}$

$\implies cos\;60=\frac{1}{2} \;\;-----(4)$

Hence, by substituting the values into the given equation we get:

From equations (1), (2), (3) and (4) we have:

$\frac{1}{2}\times\frac{\sqrt{3} }{2} -\frac{\sqrt{3} }{2} \times\frac{1}{2}$

$\implies \frac{\sqrt{3} }{4} -\frac{\sqrt{3} }{4}$

$=0$

The final answer is:

$\implies cos 60 \times cos 30 - sin 60 \times sin 30=0$

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