Question

Find the value of tan 75 degree

Answer 1

Answer:

Step-by-step explanation:

Notice that tan (75) can be written as sin(75)/cos(75) = sin(30 + 45) / cos(30 + 45)

And using a couple of trig identities, we have

[sin 30 cos 45 + sin 45 cos 30 ] / [cos 30 cos 45 -sin 30 sin 45] =

[ (1/2)(1/√2) +(1/√2))(√3/2)] / [ (√3/2) (1/√2) -(1/2) (1/√2) ] =

([1 + √3)] / [2 √2]) / ([√3 - 1] / [2 √2]) =

[ 1 + √3] / [√3 - 1] rationalizing the denominator, we have

[ 1 + √3] * [√3 + 1] / 2 =

[ 1 + √3 ] [ 1 + √3 ] / 2 =

[1 + 2√3 + 3] / 2 =

[4 + 2√3 ] / 2 =

2 + √3 ......and this is the exact value......

Answer 2

Given:

θ = 75°

To find:

value of tan θ

Solution:

We know that,

tan (A+B)=$\frac{tan A+tanB}{1-tanAtanB}$

Here,

A+B=75°=30°+45°

So, A= 30°

B=45°

Therefore, we can represent tan 75° as follows,

tan 75° = tan (30°+45°)

So, we can put the values of A and B in the above formula, to get,

tan (30°+45°)=$\frac{tan 30+tan 45}{1-tan30 tan45}$

Now, we know the value of tan 45°=1 and tan 30°=$\frac{1}{\sqrt{3} }$

Putting the values,

=$\frac{\frac{1}{\sqrt{3}}+1 }{1-(\frac{1}{\sqrt{3}})(1)}$

=$\frac{\frac{1+\sqrt{3} }{\sqrt{3} } }{\frac{\sqrt{3} -1}{\sqrt{3} } }$

=$\frac{\sqrt{3} +1}{\sqrt{3} -1}$

Hence, the value of tan 75° is $\frac{\sqrt{3} +1}{\sqrt{3} -1}$.