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Learn trigonometric function graphs with interactive visualizations. Learn sine, cosine, tangent graphs, amplitude, period, transformations & conversions.

---

## Understanding Trigonometric Function Graphs

Have you ever seen ocean waves? Their up-and-down movement forms patterns that repeat regularly. It turns out that these patterns can be modeled with trigonometric functions.

Before studying trigonometric function graphs, we need to understand angle measurement in radians. In daily life, we are accustomed to using degrees. However, in advanced mathematics, radians are more frequently used.

## Converting Degrees and Radians

One complete rotation of a circle is $$360^\circ$$ or $$2\pi$$ radians. This relationship gives us conversion formulas:

Visible text: One complete rotation of a circle is or radians. This relationship gives us conversion formulas:

Component: MathContainer
Children:

```math
180^\circ = \pi \text{ radians}
```

```math
1^\circ = \frac{\pi}{180} \text{ radians}
```

```math
1 \text{ radian} = \frac{180^\circ}{\pi} \approx 57.3^\circ
```

### Conversion Examples

Converting degrees to radians:

Component: MathContainer
Children:

```math
90^\circ = 90 \times \frac{\pi}{180} = \frac{\pi}{2} \text{ radians}
```

```math
60^\circ = 60 \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians}
```

```math
45^\circ = 45 \times \frac{\pi}{180} = \frac{\pi}{4} \text{ radians}
```

Converting radians to degrees:

Component: MathContainer
Children:

```math
\frac{\pi}{6} \text{ radians} = \frac{\pi}{6} \times \frac{180}{\pi} = 30^\circ
```

```math
\frac{3\pi}{4} \text{ radians} = \frac{3\pi}{4} \times \frac{180}{\pi} = 135^\circ
```

## What are Amplitude and Period?

Before studying trigonometric function graphs, it's important to understand two key concepts: **amplitude** and **period**.

### Amplitude

Amplitude is the maximum distance from the center line ($$x$$-axis) to the peak or trough of the graph. For functions $$y = A \sin x$$ or $$y = A \cos x$$, the amplitude is $$|A|$$.

Visible text: Amplitude is the maximum distance from the center line (-axis) to the peak or trough of the graph. For functions or , the amplitude is .

Component: LineEquation
Props:
- title: Amplitude Concept
- description: Amplitude determines the 'height' of the wave. Notice the distance from the $$x$$-axis to the peak.
  Visible text: Amplitude determines the 'height' of the wave. Notice the distance from the -axis to the peak.
- data: [
{
points: Array.from({ length: 100 }, (_, i) => {
const x = (i / 99) * 2 * Math.PI;
return { x, y: 2 * Math.sin(x), z: 0 };
}),
color: getColor("SKY"),
labels: [{ text: "Amplitude = 2", at: 25, offset: [0, 0.5, 0] }],
showPoints: false,
},
{
points: [
{ x: Math.PI / 2, y: 0, z: 0 },
{ x: Math.PI / 2, y: 2, z: 0 },
],
color: getColor("ROSE"),
showPoints: false,
},
]
- cameraPosition: [0, 0, 10]
- showZAxis: false

### Period

Period is the length of interval needed for one complete cycle. For functions $$y = \sin(Bx)$$ or $$y = \cos(Bx)$$, the period is $$\frac{2\pi}{|B|}$$.

Visible text: Period is the length of interval needed for one complete cycle. For functions or , the period is .

Component: LineEquation
Props:
- title: Period Concept
- description: Period is the horizontal distance for one complete wave.
- data: [
{
points: Array.from({ length: 100 }, (_, i) => {
const x = (i / 99) * 4 * Math.PI;
return { x, y: Math.sin(x), z: 0 };
}),
color: getColor("EMERALD"),
labels: [{ text: "Period = 2π", at: 25, offset: [0, 1.5, 0] }],
showPoints: false,
},
{
points: [
{ x: 0, y: -1.2, z: 0 },
{ x: 2 * Math.PI, y: -1.2, z: 0 },
],
color: getColor("PURPLE"),
lineWidth: 2,
showPoints: false,
cone: { position: "both" },
labels: [
{
text: "← 2π →",
at: 1,
offset: [-3, -0.5, 0],
color: getColor("PURPLE"),
},
],
},
]
- cameraPosition: [0, 0, 18]
- showZAxis: false

### General Formulas

For trigonometric functions in the form:

- $$y = A \sin(Bx)$$ and $$y = A \cos(Bx)$$:

  <MathContainer>
    
  
  ```math
  y = A \sin(Bx)
  ```

    
  
  ```math
  y = A \cos(Bx)
  ```

  </MathContainer>

- $$y = A \tan(Bx)$$:

  <MathContainer>
    
  
  ```math
  y = A \tan(Bx)
  ```

    
  
  ```math
  \text{Amplitude} = \text{undefined (infinite)}
  ```

    
  
  ```math
  \text{Period} = \frac{\pi}{|B|}
  ```

  </MathContainer>

Visible text: - and :

 <MathContainer>
 
 

 
 

 </MathContainer>

- :

 <MathContainer>
 
 

 
 

 
 

 </MathContainer>

## Sine Function Graph

The function $$y = \sin x$$ is a periodic function with period $$2\pi$$. This means its graph pattern repeats every $$2\pi \text{ interval}$$.

Visible text: The function is a periodic function with period . This means its graph pattern repeats every .

Component: LineEquation
Props:
- title: Graph of $$y = \sin x$$
  Visible text: Graph of
- description: Notice how the graph forms waves that repeat regularly.
- data: [
{
points: Array.from({ length: 100 }, (_, i) => {
const x = (i / 99) * 4 * Math.PI - 2 * Math.PI;
return { x, y: Math.sin(x), z: 0 };
}),
color: getColor("SKY"),
labels: [{ text: "y = sin x", at: 25, offset: [1.5, 0.5, 0] }],
showPoints: false,
},
]
- cameraPosition: [0, 0, 15]
- showZAxis: false

**Characteristics of** $$y = \sin x$$ **graph**:

Visible text: **Characteristics of** **graph**:

- **Period**: $$2\pi$$ (graph repeats every $$2\pi \text{ units}$$)
- **Amplitude**: $$1$$ (maximum value minus minimum value, then divided by $$2$$)
- **Domain**: All real numbers
- **Range**: $$[-1, 1]$$
- **x-intercepts**: $$x = n\pi$$ where $$n$$ is an integer
- **Maximum value**: $$1$$ at $$x = \frac{\pi}{2} + 2n\pi$$
- **Minimum value**: $$-1$$ at $$x = \frac{3\pi}{2} + 2n\pi$$

Visible text: - **Period**: (graph repeats every )
- **Amplitude**: (maximum value minus minimum value, then divided by )
- **Domain**: All real numbers
- **Range**: 
- **x-intercepts**: where is an integer
- **Maximum value**: at 
- **Minimum value**: at

## Cosine Function Graph

The function $$y = \cos x$$ has a shape similar to sine, but shifted $$\frac{\pi}{2}$$ to the left.

Visible text: The function has a shape similar to sine, but shifted to the left.

Component: LineEquation
Props:
- title: Graph of $$y = \cos x$$
  Visible text: Graph of
- description: Compare with the $$\sin$$ graph. Notice the shift.
  Visible text: Compare with the graph. Notice the shift.
- data: [
{
points: Array.from({ length: 100 }, (_, i) => {
const x = (i / 99) * 4 * Math.PI - 2 * Math.PI;
return { x, y: Math.cos(x), z: 0 };
}),
color: getColor("ROSE"),
labels: [{ text: "y = cos x", at: 25, offset: [0.3, 1.5, 0] }],
showPoints: false,
},
]
- cameraPosition: [0, 0, 15]
- showZAxis: false

**Characteristics of** $$y = \cos x$$ **graph**:

Visible text: **Characteristics of** **graph**:

- **Period**: $$2\pi$$
- **Amplitude**: $$1$$
- **Domain**: All real numbers
- **Range**: $$[-1, 1]$$
- **x-intercepts**: $$x = \frac{\pi}{2} + n\pi$$
- **Maximum value**: $$1$$ at $$x = 2n\pi$$
- **Minimum value**: $$-1$$ at $$x = \pi + 2n\pi$$

Visible text: - **Period**: 
- **Amplitude**: 
- **Domain**: All real numbers
- **Range**: 
- **x-intercepts**: 
- **Maximum value**: at 
- **Minimum value**: at

### Comparison of Sin and Cos

Component: LineEquation
Props:
- title: Comparison of $$\sin$$ and $$\cos$$ Graphs
  Visible text: Comparison of and Graphs
- description: Notice that $$\cos x = \sin(x + \frac{\pi}{2})$$.
  Visible text: Notice that .
- data: [
{
points: Array.from({ length: 100 }, (_, i) => {
const x = (i / 99) * 4 * Math.PI - 2 * Math.PI;
return { x, y: Math.sin(x), z: 0 };
}),
color: getColor("SKY"),
labels: [{ text: "y = sin x", at: 25, offset: [1.5, -1.5, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 100 }, (_, i) => {
const x = (i / 99) * 4 * Math.PI - 2 * Math.PI;
return { x, y: Math.cos(x), z: 0 };
}),
color: getColor("ROSE"),
labels: [{ text: "y = cos x", at: 75, offset: [1.5, 2.5, 0] }],
showPoints: false,
},
]
- cameraPosition: [0, 0, 15]
- showZAxis: false

## Tangent Function Graph

The function $$y = \tan x$$ differs from $$\sin$$ and $$\cos$$ because it has vertical asymptotes.

Visible text: The function differs from and because it has vertical asymptotes.

Component: LineEquation
Props:
- title: Graph of $$y = \tan x$$
  Visible text: Graph of
- description: Notice the dashed lines showing vertical asymptotes.
- data: [
// Tangent function
...Array.from({ length: 3 }, (_, periodIndex) => {
const offset = (periodIndex - 1) * Math.PI;
return {
points: Array.from({ length: 50 }, (_, i) => {
const x = offset + (i / 49) * Math.PI * 0.9 - Math.PI * 0.45;
return { x, y: Math.tan(x), z: 0 };
}),
color: getColor("EMERALD"),
labels:
periodIndex === 1
? [{ text: "y = tan x", at: 25, offset: [2, 0.5, 0] }]
: [],
showPoints: false,
};
}),
// Vertical asymptotes
...Array.from({ length: 3 }, (_, i) => {
const x = (i - 1) * Math.PI + Math.PI / 2;
return {
points: [
{ x, y: -5, z: 0 },
{ x, y: 5, z: 0 },
],
color: getColor("ROSE"),
showPoints: false,
labels:
i === 1 ? [{ text: `x = π/2`, at: 1, offset: [1, 0.5, 0] }] : [],
};
}),
]
- cameraPosition: [0, 0, 15]
- showZAxis: false

**Characteristics of** $$y = \tan x$$ **graph**:

Visible text: **Characteristics of** **graph**:

- **Period**: $$\pi$$ (shorter than $$\sin$$ and $$\cos$$)
- **Amplitude**: Undefined
- **Domain**: $$x \neq \frac{\pi}{2} + n\pi$$
- **Range**: All real numbers
- **Vertical asymptotes**: $$x = \frac{\pi}{2} + n\pi$$
- **x-intercepts**: $$x = n\pi$$

Visible text: - **Period**: (shorter than and )
- **Amplitude**: Undefined
- **Domain**: 
- **Range**: All real numbers
- **Vertical asymptotes**: 
- **x-intercepts**:

## Transformations of Trigonometric Functions

### Amplitude Changes

The function $$y = A \sin x$$ changes the amplitude to $$|A|$$.

Visible text: The function changes the amplitude to .

Component: LineEquation
Props:
- title: Effect of Amplitude
- description: Notice how the value of $$A$$ affects the wave height.
  Visible text: Notice how the value of affects the wave height.
- data: [
{
points: Array.from({ length: 100 }, (_, i) => {
const x = (i / 99) * 4 * Math.PI - 2 * Math.PI;
return { x, y: Math.sin(x), z: 0 };
}),
color: getColor("SKY"),
labels: [{ text: "y = sin x", at: 25, offset: [1.5, -2.5, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 100 }, (_, i) => {
const x = (i / 99) * 4 * Math.PI - 2 * Math.PI;
return { x, y: 2 * Math.sin(x), z: 0 };
}),
color: getColor("ROSE"),
labels: [{ text: "y = 2 sin x", at: 25, offset: [1.25, 1.5, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 100 }, (_, i) => {
const x = (i / 99) * 4 * Math.PI - 2 * Math.PI;
return { x, y: 0.5 * Math.sin(x), z: 0 };
}),
color: getColor("EMERALD"),
labels: [{ text: "y = 0.5 sin x", at: 25, offset: [1, 2.5, 0] }],
showPoints: false,
},
]
- cameraPosition: [0, 0, 15]
- showZAxis: false

### Period Changes

The function $$y = \sin(Bx)$$ changes the period to $$\frac{2\pi}{|B|}$$.

Visible text: The function changes the period to .

Component: LineEquation
Props:
- title: Effect of Period
- description: The value of $$B$$ affects how fast the function repeats.
  Visible text: The value of affects how fast the function repeats.
- data: [
{
points: Array.from({ length: 100 }, (_, i) => {
const x = (i / 99) * 4 * Math.PI - 2 * Math.PI;
return { x, y: Math.sin(x), z: 0 };
}),
color: getColor("SKY"),
labels: [{ text: "y = sin x", at: 25, offset: [1.5, -2.5, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 100 }, (_, i) => {
const x = (i / 99) * 4 * Math.PI - 2 * Math.PI;
return { x, y: Math.sin(2 * x), z: 0 };
}),
color: getColor("ROSE"),
labels: [{ text: "y = sin 2x", at: 60, offset: [1.25, 1.5, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 100 }, (_, i) => {
const x = (i / 99) * 4 * Math.PI - 2 * Math.PI;
return { x, y: Math.sin(0.5 * x), z: 0 };
}),
color: getColor("EMERALD"),
labels: [{ text: "y = sin 0.5x", at: 25, offset: [1, 2.5, 0] }],
showPoints: false,
},
]
- cameraPosition: [0, 0, 15]
- showZAxis: false

### Vertical and Horizontal Shifts

General form:

```math
y = A \sin(B(x - C)) + D
```

- $$A$$: Amplitude
- $$B$$: Affects period (
  $$\text{period} = \frac{2\pi}{|B|}$$)
- $$C$$: Horizontal shift (phase)
- $$D$$: Vertical shift

Visible text: - : Amplitude
- : Affects period (
 )
- : Horizontal shift (phase)
- : Vertical shift

Notice the horizontal and vertical shifts of the graph:

Component: LineEquation
Props:
- title: Complete Transformation
- description: Graph of $$y = 2\sin(x - \frac{\pi}{4}) + 1$$ shows all
transformations.
  Visible text: Graph of shows all
transformations.
- data: [
{
points: Array.from({ length: 100 }, (_, i) => {
const x = (i / 99) * 4 * Math.PI - 2 * Math.PI;
return { x, y: Math.sin(x), z: 0 };
}),
color: getColor("SKY"),
labels: [{ text: "y = sin x (basic)", at: 25, offset: [0.5, -2, 0] }],
showPoints: false,
},
{
points: Array.from({ length: 100 }, (_, i) => {
const x = (i / 99) * 4 * Math.PI - 2 * Math.PI;
return { x, y: 2 * Math.sin(x - Math.PI / 4) + 1, z: 0 };
}),
color: getColor("ROSE"),
labels: [
{ text: "y = 2sin(x - π/4) + 1", at: 25, offset: [0.5, 1.5, 0] },
],
showPoints: false,
},
]
- cameraPosition: [0, 0, 15]
- showZAxis: false

## Exercises

1. Convert the following angles:

   - $$120^\circ$$ to radians
   - $$\frac{5\pi}{6}$$ radians to degrees

2. Determine the period and amplitude of:

   - $$y = 3 \sin(2x)$$
   - $$y = -2 \cos(\frac{x}{3})$$

3. Sketch the graph of $$y = 2 \sin(x + \frac{\pi}{3}) - 1$$. Determine:

   - Amplitude
   - Period
   - Phase shift
   - Vertical shift

4. If tidal height is modeled by $$h(t) = 2 \sin(\frac{\pi t}{6}) + 5 \text{ meters}$$, where $$t$$ is measured in hours:

   - What are the maximum and minimum water heights?
   - What is the tidal period?

5. Determine the equation of a trigonometric function that has:

   - Amplitude $$3$$
   - Period $$\pi$$
   - Shifted $$\frac{\pi}{4}$$ to the right
   - Shifted $$2 \text{ units}$$ up

Visible text: 1. Convert the following angles:

 - to radians
 - radians to degrees

2. Determine the period and amplitude of:

 - 
 - 

3. Sketch the graph of . Determine:

 - Amplitude
 - Period
 - Phase shift
 - Vertical shift

4. If tidal height is modeled by , where is measured in hours:

 - What are the maximum and minimum water heights?
 - What is the tidal period?

5. Determine the equation of a trigonometric function that has:

 - Amplitude 
 - Period 
 - Shifted to the right
 - Shifted up

### Answer Key

1. Angle conversion:

   - $$120^\circ = 120 \times \frac{\pi}{180} = \frac{2\pi}{3}$$ radians
   - $$\frac{5\pi}{6} = \frac{5\pi}{6} \times \frac{180}{\pi} = 150^\circ$$

2. Period and amplitude:

   - $$y = 3 \sin(2x)$$: amplitude $$3$$, period $$\frac{2\pi}{2} = \pi$$
   - $$y = -2 \cos(\frac{x}{3})$$: amplitude $$2$$, period $$\frac{2\pi}{1/3} = 6\pi$$

3. For $$y = 2 \sin(x + \frac{\pi}{3}) - 1$$:

   - Amplitude: $$2$$
   - Period: $$2\pi$$
   - Phase shift: $$\frac{\pi}{3}$$ to the left
   - Vertical shift: $$1 \text{ unit}$$ down

4. For $$h(t) = 2 \sin(\frac{\pi t}{6}) + 5$$:

   - Maximum height: $$5 + 2 = 7 \text{ meters}$$
   - Minimum height: $$5 - 2 = 3 \text{ meters}$$
   - Period: $$\frac{2\pi}{\pi/6} = 12 \text{ hours}$$

5. Equation that satisfies the requirements:

   
   
   ```math
   y = 3 \sin(2(x - \frac{\pi}{4})) + 2
   ```

   or

   
   
   ```math
   y = 3 \sin(2x - \frac{\pi}{2}) + 2
   ```

Visible text: 1. Angle conversion:

 - radians
 - 

2. Period and amplitude:

 - : amplitude , period 
 - : amplitude , period 

3. For :

 - Amplitude: 
 - Period: 
 - Phase shift: to the left
 - Vertical shift: down

4. For :

 - Maximum height: 
 - Minimum height: 
 - Period: 

5. Equation that satisfies the requirements:

 
 

 or