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Understand sin, cos, and tan relationships in right triangles, then connect these basic ratios to the unit circle.

---

## Introduction to the Three Primary Trigonometric Ratios

When ancient mathematicians studied triangles, they discovered useful patterns in the ratio of sides in right triangles. There are three primary trigonometric ratios that we will learn: sine ($$\sin$$), cosine ($$\cos$$), and tangent ($$\tan$$).

Visible text: When ancient mathematicians studied triangles, they discovered useful patterns in the ratio of sides in right triangles. There are three primary trigonometric ratios that we will learn: sine (), cosine (), and tangent ().

Component: MathContainer
Children:

```math
\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}
```

```math
\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}
```

```math
\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}
```

### Understanding the Sides of a Right Triangle

Before we go further, it's important to understand the terms used in trigonometric ratios:

1. **Hypotenuse**: The longest side of a right triangle, always opposite to the right angle $$90^\circ$$.
2. **Opposite side**: The side that is opposite to the angle $$\theta$$ we are examining.
3. **Adjacent side**: The side that is adjacent to the angle $$\theta$$ we are examining, not the hypotenuse.

Visible text: 1. **Hypotenuse**: The longest side of a right triangle, always opposite to the right angle .
2. **Opposite side**: The side that is opposite to the angle we are examining.
3. **Adjacent side**: The side that is adjacent to the angle we are examining, not the hypotenuse.

Component: ContentBlock
Children:
Component: Triangle
Props:
- title: Visualization of Triangle Sides
- description: Move the slider to see how the position of sides changes with the angle.
- angle: 30
- labels: {
opposite: "Opposite Side",
adjacent: "Adjacent Side",
hypotenuse: "Hypotenuse",
}

## Sine

The sine of angle $$\theta$$ is the ratio between the length of the opposite side and the length of the hypotenuse.

Visible text: The sine of angle is the ratio between the length of the opposite side and the length of the hypotenuse.

```math
\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}
```

Component: ContentBlock
Children:
Component: Triangle
Props:
- title: Visualization of Sine ($$\sin \theta$$)
  Visible text: Visualization of Sine ()
- description: Notice how the $$\sin$$ value changes as the angle changes.
  Visible text: Notice how the value changes as the angle changes.
- angle: 30
- labels: {
opposite: "Opposite Side (determines sine value)",
adjacent: "Adjacent Side",
hypotenuse: "Hypotenuse (divisor)",
}

### Examples of Sine Values

| Angle                     | Sine Value                               | Decimal Value              |
| ------------------------- | ---------------------------------------- | -------------------------- |
| $$0^\circ$$  | $$0$$                  | $$0$$    |
| $$30^\circ$$ | $$\frac{1}{2}$$        | $$0.5$$  |
| $$45^\circ$$ | $$\frac{\sqrt{2}}{2}$$ | $$0.71$$ |
| $$60^\circ$$ | $$\frac{\sqrt{3}}{2}$$ | $$0.87$$ |
| $$90^\circ$$ | $$1$$                  | $$1$$    |

Visible text: | Angle | Sine Value | Decimal Value |
| ------------------------- | ---------------------------------------- | -------------------------- |
| | | |
| | | |
| | | |
| | | |
| | | |

## Cosine

The cosine of angle $$\theta$$ is the ratio between the length of the adjacent side and the length of the hypotenuse.

Visible text: The cosine of angle is the ratio between the length of the adjacent side and the length of the hypotenuse.

```math
\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}
```

Component: ContentBlock
Children:
Component: Triangle
Props:
- title: Visualization of Cosine ($$\cos \theta$$)
  Visible text: Visualization of Cosine ()
- description: Notice how the $$\cos$$ value changes as the angle changes.
  Visible text: Notice how the value changes as the angle changes.
- angle: 60
- labels: {
opposite: "Opposite Side",
adjacent: "Adjacent Side (determines cosine value)",
hypotenuse: "Hypotenuse (divisor)",
}

### Examples of Cosine Values

| Angle                     | Cosine Value                             | Decimal Value              |
| ------------------------- | ---------------------------------------- | -------------------------- |
| $$0^\circ$$  | $$1$$                  | $$1$$    |
| $$30^\circ$$ | $$\frac{\sqrt{3}}{2}$$ | $$0.87$$ |
| $$45^\circ$$ | $$\frac{\sqrt{2}}{2}$$ | $$0.71$$ |
| $$60^\circ$$ | $$\frac{1}{2}$$        | $$0.5$$  |
| $$90^\circ$$ | $$0$$                  | $$0$$    |

Visible text: | Angle | Cosine Value | Decimal Value |
| ------------------------- | ---------------------------------------- | -------------------------- |
| | | |
| | | |
| | | |
| | | |
| | | |

## Tangent

The tangent of angle $$\theta$$ is the ratio between the length of the opposite side and the length of the adjacent side. It can also be calculated as the ratio between the sine and cosine of the same angle.

Visible text: The tangent of angle is the ratio between the length of the opposite side and the length of the adjacent side. It can also be calculated as the ratio between the sine and cosine of the same angle.

```math
\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{\sin \theta}{\cos \theta}
```

Component: ContentBlock
Children:
Component: Triangle
Props:
- title: Visualization of Tangent ($$\tan \theta$$)
  Visible text: Visualization of Tangent ()
- description: Notice how the $$\tan$$ value changes as the angle changes.
  Visible text: Notice how the value changes as the angle changes.
- angle: 45
- labels: {
opposite: "Opposite Side (numerator)",
adjacent: "Adjacent Side (denominator)",
hypotenuse: "Hypotenuse",
}

### Examples of Tangent Values

| Angle                     | Tangent Value                            | Decimal Value              |
| ------------------------- | ---------------------------------------- | -------------------------- |
| $$0^\circ$$  | $$0$$                  | $$0$$    |
| $$30^\circ$$ | $$\frac{1}{\sqrt{3}}$$ | $$0.58$$ |
| $$45^\circ$$ | $$1$$                  | $$1$$    |
| $$60^\circ$$ | $$\sqrt{3}$$           | $$1.73$$ |
| $$90^\circ$$ | Undefined                                | Undefined                  |

Visible text: | Angle | Tangent Value | Decimal Value |
| ------------------------- | ---------------------------------------- | -------------------------- |
| | | |
| | | |
| | | |
| | | |
| | Undefined | Undefined |

## Relationship between Sin, Cos, and Tan in the Unit Circle

To understand how these trigonometric ratios work for all angles, we can use the concept of the unit circle (a circle with radius $$1$$).

Visible text: To understand how these trigonometric ratios work for all angles, we can use the concept of the unit circle (a circle with radius ).

Component: UnitCircle
Props:
- title: Unit Circle and Trigonometric Ratios
- description: Move the slider to see how the values of $$\sin$$, $$\cos$$, and $$\tan$$ change on the unit circle.
  Visible text: Move the slider to see how the values of , , and change on the unit circle.
- angle: 45

In the unit circle:

- The $$x$$-coordinate on the unit circle is $$\cos \theta$$.
- The $$y$$-coordinate on the unit circle is $$\sin \theta$$.
- $$\tan \theta$$ is the slope of the line from the center to the point on the unit circle.

Visible text: - The -coordinate on the unit circle is .
- The -coordinate on the unit circle is .
- is the slope of the line from the center to the point on the unit circle.

## Relationships Between the Three Trigonometric Ratios

These three trigonometric ratios are related by the following formulas:

Component: MathContainer
Children:

```math
\tan \theta = \frac{\sin \theta}{\cos \theta}
```

```math
\sin^2 \theta + \cos^2 \theta = 1
```

## Exercise

Consider the following triangle with a $$30^\circ$$ angle:

Visible text: Consider the following triangle with a angle:

Component: Triangle
Props:
- title: Triangle with $$30^\circ$$ Angle
  Visible text: Triangle with Angle
- description: Right triangle with a $$30^\circ$$ angle
  Visible text: Right triangle with a angle
- angle: 30
- labels: {
opposite: "Opposite Side",
adjacent: "Adjacent Side",
hypotenuse: "Hypotenuse",
}

### Check Your Answer

If the length of the hypotenuse is $$1$$, then:

Visible text: If the length of the hypotenuse is , then:

- The value of $$\sin 30^\circ$$ is the opposite-side length, $$0.5$$
- The value of $$\cos 30^\circ$$ is the adjacent-side length, $$0.87$$
- The value of $$\tan 30^\circ = \frac{\sin 30^\circ}{\cos 30^\circ} = \frac{0.5}{0.87} = 0.58$$

Visible text: - The value of is the opposite-side length, 
- The value of is the adjacent-side length, 
- The value of