MathIsimple

Trigonometry

Master trigonometric functions, identities, and formulas. From basic ratios to advanced transformations.

Trigonometry Calculator

Trigonometric Functions

The Sine Function: y = sin(x)

y = sin(x)

xyπ2π1-1
Period: 2π | Amplitude: 1 | Range: [-1, 1]

Definition

The sine function relates the angle in a right triangle to the ratio of the opposite side to the hypotenuse. In the unit circle, sin(θ) is the y-coordinate of the point on the circle.

Definition Formula

sin⁡θ=oppositehypotenuse\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}

Period

2π

Amplitude

1

Range

[-1, 1]

Zeros

x = nπ

Key Properties

  • Odd function: sin(-x) = -sin(x)
  • Maximum: sin(x) = 1 when x = π/2 + 2nπ
  • Minimum: sin(x) = -1 when x = -π/2 + 2nπ
  • Derivative: d/dx[sin(x)] = cos(x)
  • Integral: ∫sin(x)dx = -cos(x) + C

Transformations

y = A·sin(Bx + C) + D where A = amplitude, 2π/B = period, -C/B = phase shift, D = vertical shift

The Cosine Function: y = cos(x)

y = cos(x)

xyπ2π1-1
Period: 2π | Amplitude: 1 | Range: [-1, 1]

Definition

The cosine function relates the angle to the ratio of the adjacent side to the hypotenuse. In the unit circle, cos(θ) is the x-coordinate of the point on the circle.

Definition Formula

cos⁡θ=adjacenthypotenuse\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}

Period

2π

Amplitude

1

Range

[-1, 1]

Zeros

x = π/2 + nπ

Key Properties

  • Even function: cos(-x) = cos(x)
  • Maximum: cos(x) = 1 when x = 2nπ
  • Minimum: cos(x) = -1 when x = π + 2nπ
  • Derivative: d/dx[cos(x)] = -sin(x)
  • Integral: ∫cos(x)dx = sin(x) + C
  • Relationship: cos(x) = sin(x + π/2)

Key Identity

The Pythagorean identity: sin²(x) + cos²(x) = 1 — the most fundamental trig identity

The Tangent Function: y = tan(x)

y = tan(x)

xy-π/2π/2
Period: π | Range: (-∞, +∞) | Asymptotes at x = π/2 + nπ

Definition

The tangent function is the ratio of sine to cosine. It represents the slope of the line from the origin to the point on the unit circle, or opposite/adjacent in a right triangle.

Definition Formula

tan⁡θ=sin⁡θcos⁡θ=oppositeadjacent\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{\text{opposite}}{\text{adjacent}}

Period

π

Amplitude

None

Range

(-∞, +∞)

Asymptotes

x = π/2 + nπ

Key Properties

  • Odd function: tan(-x) = -tan(x)
  • Zeros: tan(x) = 0 when x = nπ
  • Undefined: at x = π/2 + nπ (where cos(x) = 0)
  • Derivative: d/dx[tan(x)] = sec²(x)
  • Integral: ∫tan(x)dx = -ln|cos(x)| + C

Related Identity

1 + tan²(x) = sec²(x) — derived from dividing sin²(x) + cos²(x) = 1 by cos²(x)

Core Concepts

Basic Trigonometric Functions

The six trigonometric functions relate angles to ratios of sides in a right triangle.

Primary Functions

Sine

sin⁡θ=oppositehypotenuse\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}

Range: [-1, 1]

Cosine

cos⁡θ=adjacenthypotenuse\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}

Range: [-1, 1]

Tangent

tan⁡θ=oppositeadjacent=sin⁡θcos⁡θ\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin\theta}{\cos\theta}

Range: (-∞, +∞)

Reciprocal Functions

Cosecant

csc⁡θ=1sin⁡θ\csc\theta = \frac{1}{\sin\theta}

Secant

sec⁡θ=1cos⁡θ\sec\theta = \frac{1}{\cos\theta}

Cotangent

cot⁡θ=1tan⁡θ=cos⁡θsin⁡θ\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}

Pythagorean Identities

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta
1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta

Special Angles

θsin(θ)cos(θ)tan(θ)
0°010
30°1/2√3/2√3/3
45°√2/2√2/21
60°√3/21/2√3
90°10undefined

Example

Problem: In a right triangle, the opposite side is 3 and hypotenuse is 5. Find sin(θ), cos(θ), and tan(θ).

Solution: sin(θ) = 3/5 = 0.6, adjacent = √(25-9) = 4, cos(θ) = 4/5 = 0.8, tan(θ) = 3/4 = 0.75

Sum and Difference Identities

These formulas express the trig functions of sums or differences of angles in terms of functions of individual angles.

sin(α + β)

sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta

sin(α - β)

sin⁡(α−β)=sin⁡αcos⁡β−cos⁡αsin⁡β\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta

cos(α + β)

cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta

cos(α - β)

cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta

tan(α + β)

tan⁡(α+β)=tan⁡α+tan⁡β1−tan⁡αtan⁡β\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}

tan(α - β)

tan⁡(α−β)=tan⁡α−tan⁡β1+tan⁡αtan⁡β\tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta}

Example

Problem: Find the exact value of sin(75°).

Solution: sin(75°) = sin(45° + 30°) = sin45°cos30° + cos45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4

Double Angle Formulas

Express trig functions of 2θ in terms of functions of θ.

Double Angle Formulas

sin(2θ)

sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin\theta\cos\theta

cos(2θ) - Form 1

cos⁡(2θ)=cos⁡2θ−sin⁡2θ\cos(2\theta) = \cos^2\theta - \sin^2\theta

cos(2θ) - Form 2

cos⁡(2θ)=2cos⁡2θ−1\cos(2\theta) = 2\cos^2\theta - 1

cos(2θ) - Form 3

cos⁡(2θ)=1−2sin⁡2θ\cos(2\theta) = 1 - 2\sin^2\theta

tan(2θ)

tan⁡(2θ)=2tan⁡θ1−tan⁡2θ\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}

Half Angle Formulas

sin(θ/2)

sin⁡θ2=±1−cos⁡θ2\sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}

cos(θ/2)

cos⁡θ2=±1+cos⁡θ2\cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}}

tan(θ/2)

tan⁡θ2=sin⁡θ1+cos⁡θ=1−cos⁡θsin⁡θ\tan\frac{\theta}{2} = \frac{\sin\theta}{1 + \cos\theta} = \frac{1 - \cos\theta}{\sin\theta}

Example

Problem: If sin(θ) = 3/5 and θ is in Q1, find sin(2θ) and cos(2θ).

Solution: cos(θ) = 4/5 (Q1). sin(2θ) = 2×(3/5)×(4/5) = 24/25. cos(2θ) = (4/5)² - (3/5)² = 16/25 - 9/25 = 7/25

Reduction Formulas

Convert trig functions of any angle to functions of an acute angle (0° to 90°).

Transformsincostan
π - θ (180° - θ)+sin(θ)-cos(θ)-tan(θ)
π + θ (180° + θ)-sin(θ)-cos(θ)+tan(θ)
2π - θ (360° - θ)-sin(θ)+cos(θ)-tan(θ)
-θ-sin(θ)+cos(θ)-tan(θ)
π/2 - θ (90° - θ)+cos(θ)+sin(θ)+cot(θ)
π/2 + θ (90° + θ)+cos(θ)-sin(θ)-cot(θ)

Memory Tip: Memory tip: Odd multiples of 90° swap sin/cos; even multiples keep them. Sign follows the quadrant.

Example

Problem: Find sin(150°), cos(210°), and tan(315°).

Solution: sin(150°) = sin(180°-30°) = sin(30°) = 1/2. cos(210°) = cos(180°+30°) = -cos(30°) = -√3/2. tan(315°) = tan(360°-45°) = -tan(45°) = -1

Quick Formula Reference

Pythagorean

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1

sin(α + β)

sin⁡αcos⁡β+cos⁡αsin⁡β\sin\alpha\cos\beta + \cos\alpha\sin\beta

sin(2θ)

2sin⁡θcos⁡θ2\sin\theta\cos\theta

cos(2θ)

cos⁡2θ−sin⁡2θ\cos^2\theta - \sin^2\theta

Practice Quiz

Practice Quiz
10
Questions
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1
What is sin(30°)?
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2
Which identity is correct?
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3
Find cos(60°).
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4
sin(α + β) = ?
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5
What is sin(2θ) in terms of sinθ and cosθ?
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6
cos(2θ) can be written as:
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7
sin(180° - θ) = ?
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8
cos(180° + θ) = ?
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9
The period of y = sin(2x) is:
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10
tan(45°) = ?
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