# Nakafa Framework: LLM

URL: /en/subject/high-school/11/mathematics/function-modeling/trigonometric-function-arbitrary-angle
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/11/mathematics/function-modeling/trigonometric-function-arbitrary-angle/en.mdx

Output docs content for large language models.

---

import { UnitCircle } from "@repo/design-system/components/contents/unit-circle";
import { Triangle } from "@repo/design-system/components/contents/triangle";

export const metadata = {
  title: "Trigonometric Function of Arbitrary Angle",
  description: "Master trigonometric functions for any angle using unit circle method. Learn quadrant signs, reference angles, and solve windmill rotation problems.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/18/2025",
  subject: "Functions and Their Modeling",
};

## Understanding Angles Greater than 90°

Have you ever observed a clock? When the minute hand moves from 12 to 6, it forms a 180° angle. Even in one complete rotation, the hand forms a 360° angle.

In mathematics, we need to understand trigonometric values for angles like these. Not just limited to acute angles in right triangles.

## Unit Circle

To understand trigonometric functions of arbitrary angles, we use the unit circle. A circle with a radius of exactly 1 unit centered at point <InlineMath math="O(0,0)" />.

<UnitCircle
  title="Unit Circle Exploration"
  description="Move the slider to see how coordinates change as the angle rotates."
  angle={30}
/>

Let's understand in detail:

- Angle <InlineMath math="\theta" /> is always measured from the positive x-axis
- Positive direction is counterclockwise
- Every point on the circle has coordinates <InlineMath math="(x, y)" />

**Important definitions:**

<div className="flex flex-col gap-4">
  <BlockMath math="\sin \theta = y \text{ (vertical coordinate)}" />
  <BlockMath math="\cos \theta = x \text{ (horizontal coordinate)}" />
  <BlockMath math="\tan \theta = \frac{y}{x} = \frac{\sin \theta}{\cos \theta}, \quad x \neq 0" />
</div>

## Why Do Signs Change in Each Quadrant?

Notice that as the point moves around the circle, the x and y coordinates can be positive or negative. This is what causes the signs of trigonometric functions to change.

<Triangle
  title="Sign Change Visualization"
  description="Observe how sin, cos, and tan values change as the angle passes through each quadrant."
  angle={0}
  size={2}
  labels={{
    opposite: "Opposite Side",
    adjacent: "Adjacent Side",
    hypotenuse: "Hypotenuse",
  }}
/>

**Signs in each quadrant:**

| Quadrant | Angle Range                                | x   | y   | sin | cos | tan |
| -------- | ------------------------------------------ | --- | --- | --- | --- | --- |
| I        | <InlineMath math="0° < \theta < 90°" />    | +   | +   | +   | +   | +   |
| II       | <InlineMath math="90° < \theta < 180°" />  | -   | +   | +   | -   | -   |
| III      | <InlineMath math="180° < \theta < 270°" /> | -   | -   | -   | -   | +   |
| IV       | <InlineMath math="270° < \theta < 360°" /> | +   | -   | -   | +   | -   |

To avoid confusion, we can remember this with **"All Students Take Calculus"**. In quadrant I **A**ll are positive, in quadrant II only **S**in is positive, in quadrant III only **T**an is positive, in quadrant IV only **C**os is positive.

## Reference Angle

A reference angle is an acute angle (0° to 90°) formed between the terminal side of an angle and the nearest x-axis. This concept allows us to use trigonometric values of acute angles that we've already memorized.

<UnitCircle
  title="Understanding Reference Angle"
  description="Notice the acute angle formed with the x-axis as the angle changes."
  angle={135}
/>

**How to determine reference angle (<InlineMath math="\alpha" />):**

- Quadrant I: <InlineMath math="\alpha = \theta" />
- Quadrant II: <InlineMath math="\alpha = 180° - \theta" />
- Quadrant III: <InlineMath math="\alpha = \theta - 180°" />
- Quadrant IV: <InlineMath math="\alpha = 360° - \theta" />

## Determining Trigonometric Values

Here are systematic steps to determine trigonometric function values:

1. **Simplify the angle** (if greater than 360° or negative)
2. **Determine the quadrant** where the angle lies
3. **Calculate the reference angle**
4. **Use the reference angle value** with the appropriate sign for the quadrant

### Angle in Quadrant II

**Problem:** Determine <InlineMath math="\sin 120°" />, <InlineMath math="\cos 120°" />, and <InlineMath math="\tan 120°" />

**Solution:**

- Angle 120° lies in quadrant II (since <InlineMath math="90° < 120° < 180°" />)
- Reference angle: <InlineMath math="\alpha = 180° - 120° = 60°" />
- In quadrant II: <InlineMath math="\sin(+), \cos(-), \tan(-)" />

Using special angle values for 60°:

<div className="flex flex-col gap-4">
  <BlockMath math="\sin 120° = +\sin 60° = \frac{\sqrt{3}}{2}" />
  <BlockMath math="\cos 120° = -\cos 60° = -\frac{1}{2}" />
  <BlockMath math="\tan 120° = -\tan 60° = -\sqrt{3}" />
</div>

### Angle in Quadrant III

**Problem:** Determine trigonometric values for angle 240°

**Solution:**

- Angle 240° lies in quadrant III (since <InlineMath math="180° < 240° < 270°" />)
- Reference angle: <InlineMath math="\alpha = 240° - 180° = 60°" />
- In quadrant III: <InlineMath math="\sin(-), \cos(-), \tan(+)" />

Using special angle values for 60°:

<div className="flex flex-col gap-4">
  <BlockMath math="\sin 240° = -\sin 60° = -\frac{\sqrt{3}}{2}" />
  <BlockMath math="\cos 240° = -\cos 60° = -\frac{1}{2}" />
  <BlockMath math="\tan 240° = +\tan 60° = \sqrt{3}" />
</div>

### Angle in Quadrant IV

**Problem:** Determine trigonometric values for angle 300°

**Solution:**

- Angle 300° lies in quadrant IV (since <InlineMath math="270° < 300° < 360°" />)

- Reference angle: <InlineMath math="\alpha = 360° - 300° = 60°" />
- In quadrant IV: <InlineMath math="\sin(-), \cos(+), \tan(-)" />

Using special angle values for 60°:

<div className="flex flex-col gap-4">
  <BlockMath math="\sin 300° = -\sin 60° = -\frac{\sqrt{3}}{2}" />
  <BlockMath math="\cos 300° = +\cos 60° = \frac{1}{2}" />
  <BlockMath math="\tan 300° = -\tan 60° = -\sqrt{3}" />
</div>

## Handling Special Angles

### Negative Angles

When the angle is negative, we move clockwise. Use the properties:

- <InlineMath math="\sin(-\theta) = -\sin \theta" /> (odd function)
- <InlineMath math="\cos(-\theta) = \cos \theta" /> (even function)
- <InlineMath math="\tan(-\theta) = -\tan \theta" /> (odd function)

**Example:** <InlineMath math="\sin(-30°) = -\sin 30° = -\frac{1}{2}" />

### Angles Greater than 360°

Use the periodicity property. Subtract or add multiples of 360° until the angle is in the range of 0° to 360°.

**Example:**

- <InlineMath math="750° = 750° - 2(360°) = 750° - 720° = 30°" />
- Therefore <InlineMath math="\sin 750° = \sin 30° = \frac{1}{2}" />

## Exercises

1. Determine the values of <InlineMath math="\sin 315°" />, <InlineMath math="\cos 315°" />, and <InlineMath math="\tan 315°" />.

2. Calculate <InlineMath math="\sin(-60°) + \cos 210° - \tan(-135°)" />.

3. If <InlineMath math="\sin \theta = \frac{3}{5}" /> and <InlineMath math="\theta" /> is in quadrant II, determine <InlineMath math="\cos \theta" /> and <InlineMath math="\tan \theta" />.

4. Simplify <InlineMath math="\sin 840° \cdot \cos(-330°)" />.

5. A windmill rotates 1050° from its initial position. If the initial position of the blade is on the positive x-axis, determine the coordinates of the blade tip on the unit circle after this rotation.

### Answer Key

1. For angle 315°, we need to determine its quadrant first.

   Since <InlineMath math="270° < 315° < 360°" />, the angle is in quadrant IV.

   The reference angle is <InlineMath math="360° - 315° = 45°" />.

   <div className="flex flex-col gap-4">
     <BlockMath math="\sin 315° = -\sin 45° = -\frac{\sqrt{2}}{2}" />
     <BlockMath math="\cos 315° = +\cos 45° = \frac{\sqrt{2}}{2}" />
     <BlockMath math="\tan 315° = -\tan 45° = -1" />
   </div>

2. Let's calculate each term separately. For <InlineMath math="\sin(-60°)" />, use the odd function property.

   For <InlineMath math="\cos 210°" />, the angle is in quadrant III with reference 30°.

   For <InlineMath math="\tan(-135°)" />, first convert to positive angle.

   <div className="flex flex-col gap-4">
     <BlockMath math="\sin(-60°) = -\sin 60° = -\frac{\sqrt{3}}{2}" />
     <BlockMath math="\cos 210° = -\cos 30° = -\frac{\sqrt{3}}{2}" />
     <BlockMath math="\tan(-135°) = -\tan 135° = -(-\tan 45°) = 1" />
     <BlockMath math="\text{Result} = -\frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2} + 1 = -\sqrt{3} + 1" />
   </div>

3. Given <InlineMath math="\sin \theta = \frac{3}{5}" /> in quadrant II.

   Use the Pythagorean identity to find <InlineMath math="\cos \theta" />.

   Remember that in quadrant II, <InlineMath math="\cos" /> is negative.

   <div className="flex flex-col gap-4">
     <BlockMath math="\sin^2 \theta + \cos^2 \theta = 1" />
     <BlockMath math="\left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1" />
     <BlockMath math="\cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}" />
     <BlockMath math="\cos \theta = -\frac{4}{5} \text{ (negative in quadrant II)}" />
     <BlockMath math="\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{3/5}{-4/5} = -\frac{3}{4}" />
   </div>

4. First simplify the angles.

   <BlockMath math="840° = 840° - 2(360°) = 120°" />

   For <InlineMath math="-330°" />, add 360° to get 30°.

   <div className="flex flex-col gap-4">
     <BlockMath math="\sin 840° = \sin 120° = \sin(180° - 60°) = \sin 60° = \frac{\sqrt{3}}{2}" />
     <BlockMath math="\cos(-330°) = \cos 330° = \cos(360° - 30°) = \cos 30° = \frac{\sqrt{3}}{2}" />
     <BlockMath math="\sin 840° \cdot \cos(-330°) = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{3}{4}" />
   </div>

5. Angle 1050° needs to be simplified first.

   <BlockMath math="1050° = 1050° - 2(360°) = 1050° - 720° = 330°" />

   Angle 330° is in quadrant IV with reference angle 30°.

   <div className="flex flex-col gap-4">
     <BlockMath math="x = \cos 330° = \cos 30° = \frac{\sqrt{3}}{2}" />
     <BlockMath math="y = \sin 330° = -\sin 30° = -\frac{1}{2}" />
     <BlockMath math="\text{Blade tip coordinates: } \left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)" />
   </div>
