Quiz 1 Topics Notes

Converting radians and degrees

Key ideas: radians and degrees are two ways to measure angles; a full circle is equivalent to both 2\pi\text{ radians} = 360^{\circ}.
Conversions:
- Degrees to radians: \theta{\text{rad}} = \theta{\text{deg}} \cdot \frac{\pi}{180}
- Radians to degrees: \theta{\text{deg}} = \theta{\text{rad}} \cdot \frac{180}{\pi}
Common angle values (in both units):
- 0^{\circ} = 0 rad
- \frac{π}{6} = 30^{\circ}
- \frac{π}{4} = 45^{\circ}
- \frac{π}{3} = 60^{\circ}
- \frac{π}{2} = 90^{\circ}
Examples:
- Convert 60^{\circ} to radians: 60^{\circ} = 60 \cdot \frac{\pi}{180} = \frac{\pi}{3} \text{ rad}
- Convert \frac{\pi}{6} \text{ rad} to degrees: \frac{\pi}{6} \cdot \frac{180}{\pi} = 30^{\circ}

Note: These identities will NOT be given to you on the quiz.
Reciprocal identities:
- \csc \theta = \frac{1}{\sin \theta}
- \sec \theta = \frac{1}{\cos \theta}
- \cot \theta = \frac{1}{\tan \theta}
- Corresponding inverses:
- \sin \theta = \frac{1}{\csc \theta}
- \cos \theta = \frac{1}{\sec \theta}
- \tan \theta = \frac{1}{\cot \theta}
Quotient identities:
- \tan \theta = \frac{\sin \theta}{\cos \theta}
- \cot \theta = \frac{\cos \theta}{\sin \theta}

These identities WILL be given to you.
Core identity:
- \sin^2 \theta + \cos^2 \theta = 1
Other standard identities:
- 1 + \tan^2 \theta = \sec^2 \theta
- 1 + \cot^2 \theta = \csc^2 \theta
Derived forms from the core identity:
- \sin^2 \theta = 1 - \cos^2 \theta
- \cos^2 \theta = 1 - \sin^2 \theta
- \tan^2 \theta = \sec^2 \theta - 1
- \cot^2 \theta = \csc^2 \theta - 1
Significance: These identities provide relationships between sine, cosine, and the tangent/cotangent families; they are essential for simplifying expressions and solving trig equations.

Definitions (based on a right triangle with angle (\theta)):
- \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}
- \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}
- \tan \theta = \frac{\text{opposite}}{\text{adjacent}}
Steps to evaluate:
- Identify the angle of interest and the lengths of the opposite, adjacent, and hypotenuse sides.
- Use the definitions above to compute the primary values (sine, cosine, tangent).
- Optionally compute the reciprocal functions as needed:
- \csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta}
Example: For a right triangle with legs 3 (opposite) and 4 (adjacent), hypotenuse 5:
- \sin \theta = \frac{3}{5}, \quad \cos \theta = \frac{4}{5}, \quad \tan \theta = \frac{3}{4}
- \csc \theta = \frac{5}{3}, \quad \sec \theta = \frac{5}{4}, \quad \cot \theta = \frac{4}{3}
Practical tips:
- When needed, use the Pythagorean theorem to find a missing side: if two sides a and b are known, the hypotenuse is c = \sqrt{a^2 + b^2}.
- Be mindful of angle location (quadrant) for sign considerations when the angle is not restricted to the first quadrant.
Real-world relevance: Core tool in physics, engineering, and computer graphics for modeling angles and lengths.