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1. Trigonometric Ratios of Negative Angles (-θ): sin(-θ) = -sinθ cosec(-θ) = -cosecθ cos(-θ) = cosθ sec(-θ) = secθ tan(-θ) = -tanθ cot(-θ) = -cotθ 2. Trigonometric Ratios of (90-θ): sin(90-θ) = cosθ cosec(90-θ) = secθ cos(90-θ) = sinθ sec(90-θ) = cosecθ tan(90-θ) = cotθ cot(90-θ) = tanθ 3. Trigonometric Ratios of (90+θ): sin(90+θ) = cosθ cosec(90+θ) = secθ cos(90+θ) = -sinθ sec(90+θ) = -cosecθ tan(90+θ) = -cotθ cot(90+θ) = -tanθ 4. Trigonometric Ratios of (180-θ): sin(180-θ) = sinθ cosec(180-θ) = cosecθ cos(180-θ) = -cosθ sec(180-θ) = -secθ tan(180-θ) = -tanθ cot(180-θ) = -cotθ 5. Trigonometric Ratios of (180+θ): sin(180+θ) = -sinθ cosec(180+θ) = -cosecθ cos(180+θ) = -cosθ sec(180+θ) = -secθ tan(180+θ) = tanθ cot(180+θ) = cotθ 6. Trigonometric Ratios of (270-θ): sin(270-θ) = -cosθ cosec(270-θ) = -secθ cos(270-θ) = -sinθ sec(270-θ) = -cosecθ tan(270-θ) = cotθ cot(270-θ) = tanθ 7. Trigonometric Ratios of (270+θ): sin(270+θ) = -cosθ cosec(270+θ) = -secθ cos(270+θ) = sinθ sec(270+θ) = cosecθ tan(270+θ) = -cotθ cot(270+θ) = -tanθ 8. Trigonometric Ratios of (360-θ): sin(360-θ) = -sinθ cosec(360-θ) = -cosecθ cos(360-θ) = cosθ sec(360-θ) = secθ tan(360-θ) = -tanθ cot(360-θ) = -cotθ 9. Trigonometric Ratios of (360+θ): sin(360+θ) = sinθ cosec(360+θ) = cosecθ cos(360+θ) = cosθ sec(360+θ) = secθ tan(360+θ) = tanθ cot(360+θ) = cotθ |
In trigonometry, allied angles are angles that are related to each other by being multiples of 90° (or π/2 radians) plus or minus another angle. Understanding the relationships between trigonometric functions of allied angles is crucial for simplifying expressions, solving equations, and extending our understanding of trigonometric behavior beyond acute angles. This guide provides a comprehensive overview of allied angle formulas, empowering you to navigate the trigonometric world with confidence.
Understanding Allied Angles: The Concept
Allied angles are angles like 90° ± θ, 180° ± θ, 270° ± θ, and 360° ± θ (or their radian equivalents). These angles are related to a reference angle θ. The formulas for allied angles express the trigonometric functions of these angles in terms of the trigonometric functions of the reference angle θ.
The Rules for Allied Angles:
There's a simple set of rules to remember when dealing with allied angles:
Even Multiples of 90° (180°, 360°): When the angle is an even multiple of 90° (or π radians), the trigonometric function remains the same. For example, sin(180° - θ) will involve sin θ.
Odd Multiples of 90° (90°, 270°): When the angle is an odd multiple of 90° (or π/2 radians), the trigonometric function changes. Specifically:
Sign Determination: The sign (+ or -) of the result depends on the quadrant in which the allied angle lies. Use the "All Students Take Calculus" (ASTC) rule to remember which functions are positive in each quadrant:
Key Allied Angle Formulas:
Here's a table summarizing the most common allied angle formulas:
| Angle | sin | cos | tan |
|---|---|---|---|
| 90° - θ | cos θ | sin θ | cot θ |
| 90° + θ | cos θ | -sin θ | -cot θ |
| 180° - θ | sin θ | -cos θ | -tan θ |
| 180° + θ | -sin θ | -cos θ | tan θ |
| 270° - θ | -cos θ | -sin θ | cot θ |
| 270° + θ | -cos θ | sin θ | -cot θ |
| 360° - θ | -sin θ | cos θ | -tan θ |
| 360° + θ | sin θ | cos θ | tan θ |
Illustrative Examples: Putting Formulas into Practice
Example 1: Find sin 120°.
Example 2: Find cos 210°.
Example 3: Find tan 300°.
Example 4: Find cot 135°.
Why This Article Matters:
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Applications of Allied Angle Formulas:
Allied angle formulas are essential for:
Conclusion: Mastering Allied Angles
Understanding allied angle formulas is a crucial step in mastering trigonometry. This guide provides a valuable resource for learning and applying these essential formulas. By understanding the rules and practicing with examples, you'll be better equipped to tackle a wider range of trigonometric problems.
Call to Action:
Bookmark this page for quick reference and share it with others who might find it helpful. Practice applying these formulas to different angle values to reinforce your understanding. The more you work with these concepts, the more confident you'll become in navigating the trigonometric world.