Question 1

What is the tangent function in trigonometry?

Accepted Answer

The tangent function (tan) is one of the six trigonometric functions. It's defined as the ratio of the opposite side to the adjacent side in a right triangle: tan(θ) = opposite/adjacent. It can also be expressed as tan(θ) = sin(θ)/cos(θ). The tangent function is periodic with period π (180°).

Question 2

How do you calculate tangent of an angle?

Accepted Answer

To calculate tangent: 1) Convert the angle to radians if given in degrees (multiply by π/180). 2) Apply the tangent function: tan(angle). 3) For special angles, use exact values: tan(0°) = 0, tan(30°) = 1/√3, tan(45°) = 1, tan(60°) = √3, tan(90°) = undefined. 4) For other angles, use a calculator or trigonometric tables.

Question 3

What are the special tangent values?

Accepted Answer

Special tangent values include: tan(0°) = 0, tan(30°) = 1/√3 ≈ 0.577, tan(45°) = 1, tan(60°) = √3 ≈ 1.732, tan(90°) = undefined (±∞), tan(120°) = -√3 ≈ -1.732, tan(135°) = -1, tan(150°) = -1/√3 ≈ -0.577, tan(180°) = 0. These values repeat every 180° due to the periodicity of tangent.

Question 4

What is the domain and range of tangent?

Accepted Answer

Domain: All real numbers except where cos(θ) = 0 (i.e., θ ≠ π/2 + nπ, where n is any integer). Range: All real numbers (-∞, +∞). The tangent function is undefined at θ = 90°, 270°, etc., where it approaches ±∞. The function is continuous everywhere except at its vertical asymptotes.

Question 5

How do you find tangent using sine and cosine?

Accepted Answer

The tangent function can be calculated using the identity: tan(θ) = sin(θ)/cos(θ). This is useful when you know the sine and cosine values. For example, if sin(30°) = 1/2 and cos(30°) = √3/2, then tan(30°) = (1/2)/(√3/2) = 1/√3. This identity is fundamental in trigonometry.

Question 6

What is the inverse tangent function?

Accepted Answer

The inverse tangent function (arctangent or tan⁻¹) returns the angle whose tangent is a given value. It's denoted as arctan(x) or tan⁻¹(x). The range of arctangent is (-π/2, π/2) or (-90°, 90°). For example, arctan(1) = π/4 = 45°, and arctan(0) = 0°.

Question 7

How do you convert between degrees and radians for tangent?

Accepted Answer

To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π. For tangent calculations: tan(degrees) = tan(degrees × π/180). Example: tan(45°) = tan(45 × π/180) = tan(π/4) = 1. The calculator handles this conversion automatically.

Question 8

What are the properties of the tangent function?

Accepted Answer

Key properties: 1) Period: π (180°), 2) Domain: All reals except π/2 + nπ, 3) Range: All reals, 4) Odd function: tan(-θ) = -tan(θ), 5) Vertical asymptotes at θ = π/2 + nπ, 6) Zero crossings at θ = nπ, 7) Increasing function in each period, 8) Symmetric about the origin.

Question 9

How is tangent used in real-world applications?

Accepted Answer

Tangent is used in: 1) Engineering: slope calculations, angle measurements, 2) Physics: projectile motion, wave analysis, 3) Architecture: roof pitch, structural analysis, 4) Navigation: bearing calculations, 5) Computer graphics: 3D rotations, lighting calculations, 6) Surveying: height and distance measurements, 7) Astronomy: celestial navigation.

Question 10

How do you solve tangent equations?

Accepted Answer

To solve tan(θ) = k: 1) Find the principal solution: θ = arctan(k), 2) Add multiples of the period: θ = arctan(k) + nπ, where n is any integer, 3) Check the domain restrictions, 4) Verify solutions by substitution. Example: tan(θ) = 1 has solutions θ = π/4 + nπ (45° + n×180°).

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