Question

Find the value of tan15​

Answer 1

Answer:

Step-by-step explanation:

Tan 15° = Tan(45 – 30)°

By the trigonometry formula, we know,

Tan (A – B) = (Tan A – Tan B) /(1 + Tan A Tan B)

Therefore, we can write,

tan(45 – 30)° = tan 45° – tan 30°/1+tan 45° tan 30°

Now putting the values of tan 45° and tan 30° from the table we get;

tan(45 – 30)° = (1 – 1/√3)/ (1 + 1.1/√3)

tan (15°) = √3 – 1/ √3 + 1

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Similarly, we can also find the value of tangent 15 degrees, by knowing the value of sin 15 and cos 15 degrees.

Tan (15°) = sin 15/cos 15

Tan 15° = sin 15/cos 15

Sin 15° = sin (45 – 30)° and cos 15 = cos (45 – 30)°

∴ tan (15°) =  sin (45 – 30)° /cos (45 – 30)°

From the trigonometry formulas, we know,

sin(A – B) = sin A cos B – cos A sin B

and cos (A – B) = cos A cos B + sin A sin B

Therefore,

tan (15°)= (sin 45° cos 30° – cos 45° sin 30°)/ (cos 45° cos 30° + sin 45° sin 30°)

Putting the values of sin 30°, sin 45°, cos 30° and cos 45°, we get,

tan 15° = [(1/√2).(√3/2) – (1/√2).(½)] / [(1/√2).(√3/2) + (1/√2).(½)]

Solving the above equation we have,

tan 15° = √3 – 1/ √3 + 1

Hence, the value of tan (15°) is √3 – 1/√3 + 1.

For Further simplifying, we can rationalize and get 2-√3

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Answer 2

SOLUTION:-

First, we have to write the given angle 15° in terms of  sum or difference of two standard angles. 

So, we have 15°  =  45° - 30°

tan15°  =  tan (45° - 30°)

tan15°  =  [tan45° -  tan30°] / [1 + tan45° tan30°]

Using the above trigonometric ratio table, we have

tan15°  =  [1 - 1/√3] / [1 + 1x1/√3]

tan15°  =  [(√3 - 1)/√3]  /  [(√3 + 1)/√3]

tan15°  =  [(√3 - 1)/√3]  x  [(√3/(√3 + 1)]

tan15°  =  (√3 - 1) / (√3 + 1)

By rationalizing the denominator, we get 

tan15°  =  2 - √3

Hence, the value of tan15° is equal to 2 - √3

MORE TO KNOW

•sin (A + B)  =  sinA cosB + cosA sinB

•sin (A - B)  =  sinA cosB - cosA sinB

•cos (A + B)  =  cosA cosB - sinA cosB

•cos (A - B)  =  cosA cosB + sinA cosB

•tan (A + B)  =  [tanA + tanB] / [1 - tanA tanB]

•tan (A - B)  =  [tanA - tanB] / [1 + tanA tanB]