Question

3 tan 60 deegrees + 4 cos 90 deegrees

Answer 1

Answer:

Derivation to Find the Value of Tan 60 Degrees



Tan 60 Degrees Value and Derivation

To find the value of tan 60 degrees geometrically, consider an equilateral triangle ABC since each of an angle in an equilateral triangle is 600.

Therefore, ∠A = ∠B = ∠C = 60°

Draw a perpendicular line AD from A to BC.

Now consider the triangle, ABD and ADC,

We have, ∠ ADB = ∠ADC= 90° and

∠ ABD = ∠ACD= 60°

Therefore, AD=AD

According to AAS Congruency,

Δ ABD ≅ Δ ACD

From this, we can say

BD = DC

Let us take, AB = BC =2a

Then, BD= ½ (BC) =½ (2a) =a

By using Pythagoras theorem,

AB2 = AD2– BD2

AD2= AB2-BD2

AD2 =(2a)2 – a2

AD2 = 4a2-a2

AD2 = 3a2

Therefore, AD=a√3

Now in triangle ADB,

Tan 600= AD/BD

= a√3/a = √3

Therefore, tan 60 degrees exact value is given by,

Tan 600=√3

In the same way, we can derive other values of tan degrees like 0°, 30°, 45°, 90°, 180°, 270° and 360°. Below is the trigonometry table, which defines all the values of tan along with other trigonometric ratios. We can easily learn the values of other tangent degrees with the help of sine functions and cosine functions. Just knowing the value of sine functions, we will find the values of cos and tan functions. There is an easy way to remember the values of tangent functions.

Sin 0° = √(0/4)

Sin 30° = √(1/4)

Sin 45° = √(2/4)

Sin 60° = √(3/4)

Sin 90° = √(4/4)

Now simplify all the sine values obtained and put in the tabular form:

Angles (in degrees)0°30°45°60°90°Sin0½1/√2√3/21

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Now find the cosine function values. It is done as follows:

Cos 0° = Sin 90°

Cos 30° = Sin 60°

Cos 45° = sin 45°

Cos 60° = sin 30°

Cos 90° = sin 0°