Ratios & Identities: Relation between Different Systems of Measurement of Angles

Understanding the relationship between different systems of measurement of angles is crucial in trigonometry. There are primarily two systems used to measure angles: degrees and radians. Let's explore these systems and how they relate to each other.

Degrees

The degree is a measure of angle equal to 1/360 of a full rotation. The symbol for degrees is °. This system divides a circle into 360 equal parts.

Radians

A radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. There are $2\pi$ radians in a full rotation. Radians are often used in higher mathematics because they simplify many formulas.

Conversion between Degrees and Radians

To convert between degrees and radians, we use the fact that $360^\circ$ is equivalent to $2\pi$ radians. Therefore, we have the following conversion formulas:

• To convert degrees to radians: $\text{radians} = \text{degrees} \times \frac{\pi}{180^\circ}$
• To convert radians to degrees: $\text{degrees} = \text{radians} \times \frac{180^\circ}{\pi}$

Table: Conversion Factors

Degrees	Radians
$360^\circ$	$2\pi$
$180^\circ$	$\pi$
$90^\circ$	$\frac{\pi}{2}$
$1^\circ$	$\frac{\pi}{180^\circ}$
$60^\circ$	$\frac{\pi}{3}$
$45^\circ$	$\frac{\pi}{4}$
$30^\circ$	$\frac{\pi}{6}$

Examples

1. Convert $90^\circ$ to radians:

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

1. Convert $\frac{3\pi}{4}$ radians to degrees:

$$ \frac{3\pi}{4} \times \frac{180^\circ}{\pi} = 135^\circ $$

Trigonometric Ratios and Identities

Trigonometric ratios such as sine, cosine, and tangent, as well as the various identities, are the same regardless of whether angles are measured in degrees or radians. However, when using these ratios and identities in calculations, it's important to ensure that the angle measurements are consistent.

Important Trigonometric Identities

• Pythagorean Identity: $\sin^2\theta + \cos^2\theta = 1$
• Sum and Difference Formulas:
  • $\sin(\alpha \pm \beta) = \sin\alpha \cos\beta \pm \cos\alpha \sin\beta$
  • $\cos(\alpha \pm \beta) = \cos\alpha \cos\beta \mp \sin\alpha \sin\beta$
• Double Angle Formulas:
  • $\sin(2\theta) = 2\sin\theta \cos\theta$
  • $\cos(2\theta) = \cos^2\theta - \sin^2\theta$

Example Using Identities

Calculate $\sin(60^\circ)$ using the identity for $\sin(2\theta)$:

First, convert $60^\circ$ to radians:

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

Now, use the double angle formula for sine:

$$ \sin(2 \times \frac{\pi}{3}) = 2\sin(\frac{\pi}{3})\cos(\frac{\pi}{3}) $$

Since $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$ and $\cos(\frac{\pi}{3}) = \frac{1}{2}$:

$$ \sin(2 \times \frac{\pi}{3}) = 2 \times \frac{\sqrt{3}}{2} \times \frac{1}{2} = \frac{\sqrt{3}}{2} $$

Conclusion

Understanding the relationship between degrees and radians is essential for working with angles in trigonometry. Whether you're using trigonometric ratios or identities, it's important to ensure that your angle measurements are consistent throughout your calculations. Remember to use the conversion formulas to switch between degrees and radians as needed.