Problem 19 - Algebra 2 and Trigonometry

Question

a. Find the exact value of \(\cos 120^{\circ}\) by using \(\cos \left(60^{\circ}+60^{\circ}\right) .\) b. Find the exact value of \(\sin 120^{\circ}\) by using \(\cos ^{2} \theta+\sin ^{2} \theta=1\) and the value of \(\cos 120^{\circ}\) found in a. c. Find the exact value of \(\cos 165^{\circ}\) by using \(\cos \left(120^{\circ}+45^{\circ}\right)\) d. Use the value of \(\cos 165^{\circ}\) found in \(c\) to find \(\cos 345^{\circ}\) by using \(\cos \left(180^{\circ}+165^{\circ}\right)\)

Step-by-Step Solution

Answer

a. \(-\frac{1}{2}\); b. \(\frac{\sqrt{3}}{2}\); c. \(-\frac{\sqrt{2} + \sqrt{6}}{4}\); d. \(\frac{\sqrt{2} + \sqrt{6}}{4}\).

Step 1: Find \(\cos 120^{\circ}\) using the Angle Addition Formula

Use the angle addition formula for cosine: \[ \cos(a + b) = \cos a \cos b - \sin a \sin b \].Here, let \(a = b = 60^{\circ}\), we have:\[ \cos(120^{\circ}) = \cos(60^{\circ} + 60^{\circ}) = \cos 60^{\circ} \cos 60^{\circ} - \sin 60^{\circ} \sin 60^{\circ} \]Substitute known values: \(\cos 60^{\circ} = \frac{1}{2}, \sin 60^{\circ} = \frac{\sqrt{3}}{2}\):\[ \cos 120^{\circ} = \left(\frac{1}{2}\right)^2 - \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} - \frac{3}{4} = -\frac{1}{2} \].Therefore, \(\cos 120^{\circ} = -\frac{1}{2}\).

Step 2: Find \(\sin 120^{\circ}\) using the Pythagorean Identity

Using the identity \(\cos^2 \theta + \sin^2 \theta = 1\), substitute \(\theta = 120^{\circ}\) and \(\cos 120^{\circ} = -\frac{1}{2}\):\[ (-\frac{1}{2})^2 + \sin^2(120^{\circ}) = 1 \]\[ \frac{1}{4} + \sin^2(120^{\circ}) = 1 \]\[ \sin^2(120^{\circ}) = 1 - \frac{1}{4} \]\[ \sin^2(120^{\circ}) = \frac{3}{4} \]Taking the positive square root (as 120° is in the second quadrant where sine is positive):\[ \sin 120^{\circ} = \frac{\sqrt{3}}{2} \].

Step 3: Find \(\cos 165^{\circ}\) using the Angle Addition Formula

Using the angle addition formula: \[ \cos(a + b) = \cos a \cos b - \sin a \sin b \],let \(a = 120^{\circ}\), \(b = 45^{\circ}\).\[ \cos(165^{\circ}) = \cos(120^{\circ} + 45^{\circ}) \]Substitute known values: \(\cos 120^{\circ} = -\frac{1}{2}\), \(\sin 120^{\circ} = \frac{\sqrt{3}}{2}\), \(\cos 45^{\circ} = \sin 45^{\circ} = \frac{\sqrt{2}}{2}\):\[ \cos(165^{\circ}) = (-\frac{1}{2}) \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} \]\[ \cos(165^{\circ}) = -\frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} \]\[ \cos(165^{\circ}) = -\frac{\sqrt{2} + \sqrt{6}}{4} \].

Step 4: Find \(\cos 345^{\circ}\) using the Cosine of a Sum Formula

Relate \(345^{\circ}\) to known angles: \(345^{\circ} = 180^{\circ} + 165^{\circ}\).Using the cosine angle addition formula:\[ \cos (180^{\circ} + \theta) = -\cos \theta \]\[ \cos 345^{\circ} = -\cos 165^{\circ} \]From Step 3, \(\cos 165^{\circ} = -\frac{\sqrt{2} + \sqrt{6}}{4}\):\[ \cos 345^{\circ} = -\left(-\frac{\sqrt{2} + \sqrt{6}}{4}\right) = \frac{\sqrt{2} + \sqrt{6}}{4} \].

Key Concepts

Angle Addition Formula

The angle addition formula is a crucial tool in trigonometry. It helps simplify the process of finding the sine or cosine of a sum of two angles. The formulas are:

• For cosine: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)
• For sine: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)

This means you can break down a complex angle calculation into simpler pieces that you know.
In the problem, we used this formula to find \( \cos 120^{\circ} \) by setting \( a = b = 60^{\circ} \).
The steps were straightforward:

• Plug in known values: \( \cos 60^{\circ} = \frac{1}{2} \) and \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)
• Calculate: \( (\frac{1}{2})^2 - (\frac{\sqrt{3}}{2})^2 = \frac{1}{4} - \frac{3}{4} = -\frac{1}{2} \)
  Thus, \( \cos 120^{\circ} = -\frac{1}{2} \).

It's like solving a puzzle by opting for familiar and smaller pieces.

Cosine of a Sum Formula

The cosine of a sum formula, \( \cos(a + b) = \cos a \cos b - \sin a \sin b \), allows us to express the cosine of two added angles minus their sines multiplied. Here, it was used in two primary ways:
First, to find \( \cos 165^{\circ} \). We rewrote it using two known angles: \( a = 120^{\circ} \) and \( b = 45^{\circ} \).
This gave us:

• \( \cos 165^{\circ} = \cos 120^{\circ} \cos 45^{\circ} - \sin 120^{\circ} \sin 45^{\circ} \)
• Substituting values: \( -\frac{1}{2} \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} \)
• Outcome: \( -\frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} = -\frac{\sqrt{2} + \sqrt{6}}{4} \)

Secondly, for \( \cos 345^{\circ} \), by relating it to known values via subtraction or addition with 180 degrees.
Overall, this formula enables breaking down solutions due to its simplicity and versatility.

Pythagorean Identity

This identity, \( \cos^2 \theta + \sin^2 \theta = 1 \), represents a fundamental relationship between sine and cosine functions for any angle \( \theta \). It essentially displays the inherent ties they're bound to on the unit circle.
In part (b) of the exercise, it helped us calculate \( \sin 120^{\circ} \). With \( \cos 120^{\circ} = -\frac{1}{2} \), plugging into Pythagorean identity results:

• \( (-\frac{1}{2})^2 + \sin^2 120^{\circ} = 1 \)
• \( \frac{1}{4} + \sin^2 120^{\circ} = 1 \)
• Solving: \( \sin^2 120^{\circ} = \frac{3}{4} \), and since 120° is in the second quadrant (where sine is positive): \( \sin 120^{\circ} = \frac{\sqrt{3}}{2} \)

The identity serves almost like a safe formula for checking your answers and making sure they align symmetrically with the unit circle surrounding us.