Converting Degrees to Radians

Convert $45°$ to radians.

Write the conversion formula: Radians $= $ Degrees $\times \frac{\pi}{180}$ = Formula ready

Substitute the value: $45 \times \frac{\pi}{180}$ = $\frac{45\pi}{180}$

Simplify the fraction: $\frac{45\pi}{180} = \frac{\pi}{4}$ = Divide both by 45

Answer: $45° = \frac{\pi}{4}$ radians

Converting Radians to Degrees

Convert $\frac{2\pi}{3}$ radians to degrees.

Write the conversion formula: Degrees $= $ Radians $\times \frac{180}{\pi}$ = Formula ready

Substitute the value: $\frac{2\pi}{3} \times \frac{180}{\pi}$ = Set up multiplication

Cancel $\pi$ and simplify: $\frac{2 \times 180}{3} = \frac{360}{3}$ = $= 120$

Answer: $\frac{2\pi}{3}$ radians $= 120°$

Finding Radian Measure on the Unit Circle

An angle measures $\frac{5\pi}{6}$ radians. Where is this on the unit circle, and what is its degree measure?

Convert to degrees: $\frac{5\pi}{6} \times \frac{180}{\pi} = \frac{5 \times 180}{6} = 150°$ = $150°$

Locate on unit circle: $150°$ is in Quadrant II, $30°$ away from the negative $x$-axis = Second quadrant

Identify reference angle: $180° - 150° = 30°$ = Reference angle is $30°$

Answer: $\frac{5\pi}{6}$ radians $= 150°$, located in Quadrant II with a reference angle of $30°$

Mistake: Forgetting to simplify the fraction after conversion

Why: The raw calculation often gives an unsimplified result like $\frac{90\pi}{180}$.

Correct: Always simplify: $\frac{90\pi}{180} = \frac{\pi}{2}$. Look for common factors.

Mistake: Using degrees when calculator is in radian mode (or vice versa)

Why: Calculators have different modes. $\sin(90)$ in radian mode gives approximately $0.894$, not $1$.

Correct: Always check your calculator mode. $\sin(90°) = 1$ but $\sin(90 \text{ rad}) \approx 0.894$.

Mistake: Confusing $\pi$ radians with $180\pi$ degrees

Why: Mixing up the conversion direction leads to huge errors.

Correct: $\pi$ radians $= 180°$ (not $180\pi°$). Remember: $\pi \approx 3.14$ is the radian value.

Rotating Objects in Video Games

Game developers use radians to rotate characters and objects smoothly on screen.

A character turning $90°$ to face right rotates $\frac{\pi}{2}$ radians in the code.

Circular Motion in Physics

Scientists measure angular velocity in radians per second for rotating objects.

A wheel spinning at $2\pi$ radians per second completes exactly one full rotation every second.

Q: Why do we need radians when degrees work fine?

A: Radians make calculus formulas much simpler. For example, the derivative of $\sin(x)$ is $\cos(x)$ only when $x$ is in radians. With degrees, you would need an extra conversion factor.

Q: What does 1 radian look like?

A: One radian is the angle where the arc length equals the radius. It is approximately $57.3°$. Think of wrapping the radius around the circle's edge - that arc spans 1 radian.

Q: How do I remember the common conversions?

A: Start with $180° = \pi$. Then $90° = \frac{\pi}{2}$ (half), $60° = \frac{\pi}{3}$ (third), $45° = \frac{\pi}{4}$ (quarter), $30° = \frac{\pi}{6}$ (sixth).