 Use the double-angle formulas to determine (3 pts each):

a. cos 120° b. tan 300°

Let's solve the given problems using the double-angle formulas:

### Part (a): \(\cos(120^\circ)\)

We can express \(120^\circ\) in terms of the double angle as:
\[
\cos(120^\circ) = \cos(2 \times 60^\circ)
\]

Using the double-angle formula for cosine:
\[
\cos(2\theta) = 2\cos^2(\theta) - 1
\]
Let \(\theta = 60^\circ\). Then, we have:
\[
\cos(120^\circ) = 2\cos^2(60^\circ) - 1
\]
We know that \(\cos(60^\circ) = \frac{1}{2}\). Substituting this value into the equation:
\[
\cos(120^\circ) = 2\left( \frac{1}{2} \right)^2 - 1
\]
\[
\cos(120^\circ) = 2 \cdot \frac{1}{4} - 1
\]
\[
\cos(120^\circ) = \frac{1}{2} - 1
\]
\[
\cos(120^\circ) = -\frac{1}{2}
\]

### Part (b): \(\tan(300^\circ)\)

We can express \(300^\circ\) in terms of the double angle as:
\[
\tan(300^\circ) = \tan(2 \times 150^\circ)
\]

Using the double-angle formula for tangent:
\[
\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}
\]
Let \(\theta = 150^\circ\). We know that:
\[
\tan(150^\circ) = -\tan(30^\circ) = -\frac{1}{\sqrt{3}}
\]

Substituting this into the formula:
\[
\tan(300^\circ) = \frac{2 \left( -\frac{1}{\sqrt{3}} \right)}{1 - \left( -\frac{1}{\sqrt{3}} \right)^2}
\]
\[
\tan(300^\circ) = \frac{-\frac{2}{\sqrt{3}}}{1 - \frac{1}{3}}
\]
\[
\tan(300^\circ) = \frac{-\frac{2}{\sqrt{3}}}{\frac{2}{3}}
\]
\[
\tan(300^\circ) = -\frac{2}{\sqrt{3}} \cdot \frac{3}{2}
\]
\[
\tan(300^\circ) = -\sqrt{3}
\]

### Final Answers:
a. \(\cos(120^\circ) = -\frac{1}{2}\)  
b. \(\tan(300^\circ) = -\sqrt{3}\)

Would you like more details or have any questions about these calculations?

Here are 5 related questions to expand your understanding:
1. What are the double-angle formulas for sine and cosine?
2. How do you derive the double-angle formula for tangent?
3. What is the value of \(\sin(120^\circ)\) using double-angle formulas?
4. How can we express \(240^\circ\) using double-angle formulas?
5. What are the values of trigonometric functions for angles like \(150^\circ\) and \(210^\circ\)?

**Tip:** Remember that double-angle formulas are very useful for transforming expressions involving multiples of an angle into simpler forms.

Use the double-angle formulas to determine: a. cos 120° b. tan 300°

Learn how to determine cos(120°) and tan(300°) using double-angle formulas in trigonometry. This problem explains key concepts such as the double-angle formula for cosine and tangent with detailed step-by-step solutions. Ideal for students in Grades 10-12.

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