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Tan 0 Degrees is equal to zero (0). Tangent is one of the three primary trigonometric functions along with Sine and Cosine.
- Tangent Function is the ratio of the opposite side and the adjacent side of a right-angled triangle.
- It can also be expressed as the ratio of the sine function and cosine function.
- The value of Tan 0 Degrees is equal to 0.
- Tan 0 Degrees is written as Tan (0° x π/180°) in radians, i.e. Tan (0π) or Tan (0).
Trigonometry is the branch of mathematics primarily concerned with the relationship between the side lengths and the angles of a triangle. It is applicable in many fields such as surveying, geodesy, navigation, optics, acoustics, etc.
Read More: NCERT Solutions for Class 11 Mathematics Trigonometric Functions
Key Terms: Tan 0 Degrees, Tangent Function, Tan, Sine, Cosine, Trigonometric Functions, Trigonometry, Right-angled Triangle
What is Tangent?
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Tangent is a trigonometric function that is defined as the ratio of the opposite side to the adjacent side.
- It is generally expressed as tan x.
- Tangent Function is the ratio of the perpendicular to the base in a right-angled triangle.
- It is a primary trigonometric function that helps in the derivation of the Cotangent Function.
- It is also expressed as a ratio of two other primary trigonometric functions Sine and Cosine.
- Tangent is a periodic function and has a period of π/1 = π.
Tangent Function
Value of Tan 0 Degrees
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Value of Tan 0 Degrees is 0. Tan 0 Degrees can also be expressed through the equivalent measure of the given angle (0 Degrees) in radians (0...).
| Value of Tan 0 Degrees = 0 |
In obtain the value of tan 0 degrees in radians, Degrees to Radians Conversion is done with the help of the formula:
θ in Radians = θ in Degrees × (π/180°)
0 Degrees = 0° × (π/180°) rad = 0π or 0…
Therefore,
| tan 0° = tan(0) = 0 |
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Explanation for Value of Tan 0 Degrees
For tan 0 Degrees, the angle 0° lies on the positive x-axis due to which the value of tan 0° is 0. Tangent Function is a periodic function, thus, it can be expressed as
tan(0° + n × 180°), n ∈ Z
tan 0° = tan 180° = tan 360°, and so on.
Important Note: As Tangent is an odd function, the value of tan (-0°) = -tan(0°) = 0.
Trigonometric Functions Detailed Video Explanation
Derivation of Value of Tan 0 Degrees
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Value of Tan 0 Degrees is zero. The value can be derived and proved using two methods which are as follows:
- Using Trigonometric Functions
- Using Unit Circle
Tan 0 Degrees in Terms of Trigonometric Functions
(I) Tan 0 Degrees can be represented as follows in terms of trigonometry formulas:
- sin(0°)/cos(0°)
- ± sin 0°/√(1 - sin²(0°))
- ± √(1 - cos²(0°))/cos 0°
- ± 1/√(cosec²(0°) - 1)
- ± √(sec²(0°) - 1)
- 1/cot 0°
Important Note: The final value of Tan 0 Degrees is 0 as 0° lies on the positive x-axis.
(II) Tan 0 Degrees can also be expressed in terms of trigonometric identities as follows:
- cot(90° - 0°) = cot 90°
- -cot(90° + 0°) = -cot 90°
- -tan (180° - 0°) = -tan 180°
(III) Tan function is also expressed as the ratio of sine and cosine function as follows.
Tanθ = Sinθ/Cosθ
If the angle θ = 0°,
It can be written as:
Tan 0° = Sin0°/Cos0°
We know that, Sin 0° = 0 and Cos 0° = 1,
On substituting, we get
Tan0° = 0/1
Thus, Tan 0° = 0
Hence Proved.
Tan 0 Degrees Using Unit Circle
In order to find the value of tan 0 degrees using the unit circle, the following steps need to be followed:
- Draw the radius ‘r’ of the unit circle to create a 0° angle with the positive x-axis.
- The tan of 0 degrees equals the y-coordinate(0) divided by the x-coordinate (1) of the point of intersection (1, 0) of the unit circle and r.
Value of Tan 0 Degrees Using Unit Circle
Hence, the value of tan 0 Degrees will be
| Tan 0° = y/x = 0 |
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Trigonometry Ratio Table
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Trigonometry Table enlists all the values of the Tangent Function for common angles like 0°, 30°, 45°, 60°, 90°, 180°, etc along with other trigonometric ratios. The six trigonometric ratios are sine, cosine, tangent, cosecant, secant, and cotangent whose common values are listed as follows:
| Trigonometry Table | ||||||||
|---|---|---|---|---|---|---|---|---|
| Angles (In Degrees) | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
| Angles (In Radians) | 0° | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π |
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 | 0 | -1 | 0 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 | -1 | 0 | 1 |
| tan | 0 | 1/√3 | 1 | √3 | ∞ | 0 | ∞ | 0 |
| cot | ∞ | √3 | 1 | 1/√3 | 0 | ∞ | 0 | ∞ |
| cosec | ∞ | 2 | √2 | 2/√3 | 1 | ∞ | -1 | ∞ |
| sec | 1 | 2/√3 | √2 | 2 | ∞ | -1 | ∞ | 1 |
Things to Remember
- Tangent Function is a primary trigonometric ratio generally expressed as Tan x.
- It is the ratio of the opposite side to the adjacent side of a right-angled triangle.
- Tangent Function is also the ratio of Sine and Cosine Function, i.e. Tan θ = Sin θ/Cos θ.
- The exact value of Tan 0 Degrees is Zero (0).
- Value of Tan 0 Degrees is written as Tan (0π) or Tan (0) in Radians.
- The value of Tan 0 Degrees can be derived using other Trigonometric Functions and the Unit Circle.
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Previous Years’ Questions
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Sample Questions
Ques. Find the value of given expressions:
(a) 3 tan(0°)/7 tan(45°)
(b) tan 0° + cos 0°
(c) tan 0° + cot 45° (3 Marks)
Ans. The value of Tan 0 Degrees is 1.
(a) Using Trigonometry Values,
- tan(0°) = 0
- tan 45° = 1
Thus, the value of 3 tan(0°)/7 tan(45°) = (3 x 0)/(7x1) = 0.
(b) Using Trigonometry Values,
- tan 0° = 0
- cos 0° = 1
Thus, the value of tan 0° + cos 0° = 0 + 1 = 1.
(c) Using Trigonometry Values,
- tan 0° = 0
- cot 45° = 1
Thus, the value of tan 0° + cot 45° = 0 + 1 = 1
Ques. If tan θ + cot θ = 5, find the value of tan² θ + cot² θ. (3 Marks)
Ans. It is given that tan θ + cot θ = 5.
On squaring both sides, we get
tan² θ + cot² θ + 2 tan θ cot θ = 25
tan² θ + cot² θ + 2 = 25
∴ tan² θ + cot² θ = 23
Thus, the value of tan² θ + cot² θ is equal to 23 if tan θ + cot θ = 5.
Ques. What is the value of Tan 0° in terms of other trigonometric functions? (3 Marks)
Ans. The value of Tan 0° in terms of other trigonometric functions is as follows:
- sin(0°)/cos(0°)
- ± sin 0°/√(1 - sin²(0°))
- ± √(1 - cos²(0°))/cos 0°
- ± 1/√(cosec²(0°) - 1)
- ± √(sec²(0°) - 1)
- 1/cot 0°
Ques. If sec θ + tan θ = 7, then evaluate sec θ – tan θ. (3 Marks)
Ans. According to the trigonometric identities, we know that,
sec²θ – tan²θ = 1
On expanding the identity,
(sec θ + tan θ) (sec θ – tan θ) = 1
Substitute the value of sec θ + tan θ = 7 in the expression,
(7) (sec θ – tan θ) = 1
Thus, sec θ – tan θ = 1/7.
Ques. If x = p sec θ + q tan θ and y = p tan θ + q sec θ, then prove that x² – y² = p² – q². (3 Marks)
Ans. Given that, x = p sec θ + q tan θ and y = p tan θ + q sec θ.
L.H.S. = x² – y²
= (p sec θ + q tan θ)² – (p tan θ + q sec θ)²
= p² sec θ + q² tan² θ + 2 pq sec² tan² -(p² tan² θ + q² sec² θ + 2pq sec θ tan θ)
= p² sec θ + 2 tan² θ + 2pq sec θ tan θ – p² tan² θ – q² sec θ – 2pq sec θ tan θ
= p²(sec² θ – tan² θ) – q²(sec² θ – tan² θ)
= p² – q² …[As sec² θ – tan² θ = 1]
R.H.S. = p² – q²
L.H.S. = R.H.S.
Hence Proved.
Ques. What is the value of Tan 0 Degrees in terms of Sec 0°? (2 Marks)
Ans. Tangent function can be expressed in terms of the secant function using trigonometric identities as
tan 0° = √(sec²(0°) - 1)
Thus, the value of sec 0° is equal to 1.
Ques. If the value of (tan θ + cot θ) = 5, find out the value of tan2θ + cot2θ. (3 Marks)
Ans. It is given that, tan θ + cot θ = 5.
Squaring both sides we get,
tan2 θ + cot2 θ + 2 tanθ cotθ = 25
tan2θ + cot2θ + 2tanθ/tanθ = 25 (As Cotθ = 1/tanθ)
tan2θ + cot2θ + 2 =25
tan2θ + cot2θ = 25-2
tan2θ + cot2θ = 23
So, the value of tan2θ + cot2θ = 23.
Ques. If the value of sec θ + tan θ = 9, then evaluate sec θ – tan θ. (3 Marks)
Ans. According to the trigonometric identities, we know that,
We know, sec2θ – tan2θ =1
On expanding the identity, we get
(sec θ + tan θ)( sec θ – tan θ)=1
9(sec θ – tan θ)=1
sec θ – tan θ =1/9
So, the value of sec θ – tan θ = 1/9.
Ques. Prove that (sin θ + cos θ + 1) (sin θ – 1 + cos θ) . sec θ . cosec θ = 2. (3 Marks)
Ans. (sin θ + cos θ + 1) (sin θ – 1 + cos θ) . sec θ . cosec θ
= [(sin θ + cos θ) + 1] [(sin θ + cos θ) – 1] . sec θ cosec θ
= [(sin θ + cos θ)2 – (1)2] sec θ cosec θ …[\(\because\) (a + b)(a – b) = a2 – b2]
= (sin2 θ + cos2θ + 2 sin θ cos θ – 1]. sec θ cosec θ
= (1 + 2 sin θ cos θ – 1). sec θ cosecθ …[\(\because\)sin2θ + cos2θ = 1]
= (2 sin θ cos θ). 1/cosθ.1/sinθ
= 2
Hence Proved
Ques. If tan (20° – 3α) = cot( 5α – 20°), then find the value of α. (2 Marks)
Ans. Given that tan(20° – 3α) = cot(5α – 20°).
tan(20° – 3α) = tan[90° – (5α – 20°)] … [\(\because\)cot θ = tan(90° – θ)]
∴ 20° – 3α = 90° – 5α + 20°
→ -3α + 5α = 90° + 20° – 20°
→ 2α = 90° ⇒ α = 45°
Thus, the value of a is 45°.
Ques. Find the value of Tan 15°. (3 Marks)
Ans. Tan 15° = Tan (45° – 30°)
According to the trigonometric formula,
tan (A-B) = (tan A – tan B) / (1+ tan A tan B)
Substituting the values of tan 30° and tan 45°
Tan 15° = tan (45° – 30°)
= (tan45° - tan30°)/ (1+ tan45°tan30°)
= 1
Thus, the value of Tan 15° is 1.
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