Value of Tan 15 degrees: Calculation Using Sin & Cos Function

Value of Tan 15° can be found if the value of sin 15 degrees and cos 15 degrees are known. The tangent of an angle is equivalent to the ratio of sine and cosine functions of the same angle, in the right angle triangle. Trigonometry is the type of mathematics in which the relationships between the angles and the sides of the triangles are set out. There are six types of trigonometry ratios involved in mathematics. These ratios are as follows; Sine of the angle, cosine of the angle, the tangent of an angle, cosecant of angle, secant of angle and cotangent of angle. The concept of the trigonometry ratio is applicable in a right-angle triangle. The right-angle triangle contains one angle of 90 degrees.

Key terms: Trigonometry ratios, Tangent formula, Trigonometry tables.

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Value of Tan 15 Degres

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The value of tan (15°) is 0.26794919243. A right-angled triangle consists of the angles and sides that are determined by angles and sides of a right-angled triangle respectively. Thus, trigonometry can be referred to as the measurement of triangles, particularly right-angled triangles. If one of the interior angles of a triangle is 90 degrees, the triangle is right-angled. Hence, the three sides of the triangle can be written as follows:

• Hypotenuse: It is the side opposite the 90-degree angle in a right-angle triangle
• Perpendicular: It is the opposite side of any angle between base and perpendicular. This side is perpendicular to the base.
• Base: It is the side adjacent to both the unknown angle and the 90 degrees angle.

Sine of angle is the ratio of the side opposite to that angle and the hypotenuse in a right-angle triangle. The cosine of the angle is the ratio of the side adjacent to that angle and the hypotenuse of the right-angle triangle. The tangent of angle refers to the side opposite to the angle and the side adjacent to that angle. The reverse ratio of sine, cosine and tangent indicates the cosine, secant and cotangent ratio in the right-angle triangle.

• Sin A= P/H
• Cos A=B/H
• Tan A=P/B

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What is the Value of Tan 15 Degrees?

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Let’s have a look at the trigonometry table for sin, cos and tan.

• Tangent formula

Tangent A: Opposite side to angle A

Adjacent side to angle A

Read More: Trigonometry Values

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How to find the Value of Tan 15 Degrees?

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Let's take a right-angle triangle having one angle of 15 degrees. Since in a right angle one angle always remains 90 degrees.

• Sum of all the angles in a triangle = 180 degrees

⇒∠ P + ∠Q+ ∠R= 180

⇒90+15+ ∠R=180

⇒∠R=75

Tan 15 = Tan (45-30)

As per the above identity of Tan (A-B):

Tan (A-B): Tan A – Tan B

1+ (Tan A *Tan B)

Tan Q= Tan (A-B)

⇒ Tan 15= Tan (45-30)

Tan (45-30) = Tan 45- Tan 30

1+ Tan 45*Tan 30

⇒ Tan (45-30) = {1- 1/ √3} [Tan 45= 1 and Tan 30= 1/√3]

{1+ (1*1/ √3)}

⇒ Tan 15= {√3 – 1}/ {√3+1} [By rationalising the denominator]

∴ Tan 15= 2-√3

 [√3= 1.73205]

⇒Tan 15= 2-1.73205

 Tan 15= 0.267949

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Value of Tan 15 Degree with respect to Sin and Cos Function

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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

Things to Remember

• Tangent A: Opposite side to angle A = Adjacent side to angle A
• The formula of Tan (A-B) = Tan A – Tan B = 1+ (Tan A *Tan B)
• Tan30=1/ √3
• Tan45=1
• Trigonometric ratios apply only in a right-angle triangle.
• Trigonometric integrates conceptual learning, memorising and problem-solving skills of students.
• With the word trigonometry, a triangle measure word is generated

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