Problem 17 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Find the exact value of each expression. $$ \cos \left(135^{\circ}+30^{\circ}\right) $$

Step-by-Step Solution

Answer

The exact value of the expression \( \cos(135^{\circ}+30^{\circ}) \) is \( \frac{\sqrt{3}}{2} \).

Step 1: Simplify the Inner Expression

First, simplify the inner expression 135^{\circ}+30^{\circ} to reduce the cosine function to an equivalent form. Doing this gives \(\cos(165^{\circ})\).

Step 2: Use Reference Angles

The next step is to use reference angles to find a simpler, equivalent angle. Since the cosine function has a period of 360°, \(\cos(165^{\circ})\) has the same value as \(\cos(195^{\circ})\) because 195^{\circ} is 30^{\circ} more than 165^{\circ}, and adding 30^{\circ} to 165^{\circ} makes the result in the fourth quadrant. The cosine function is positive in the fourth quadrant.

Step 3: Use Known Values of Cosine

Finally, use the known value of cosine at 30°. So \(\cos(195^{\circ})\) is equivalent to \(\cos(30^{\circ})\), which is \( \frac{\sqrt{3}}{2} \).

Key Concepts

Cosine Function

The cosine function, represented as \( \cos \theta \), is a fundamental concept in trigonometry that deals with the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Its values range between -1 and 1 for all angles. Cosine is even and periodic, which means it repeats its values in regular intervals of 360°.

• This periodic nature is handy for simplifying angle calculations, particularly by translating angles more than 360° back into primary positions within a circle.
• The cosine of negative angles illustrates its even nature, implying that \( \cos(-\theta) = \cos(\theta) \).

It's also crucial to remember that the cosine function is positive in the first and fourth quadrants of the unit circle, hence altering angles into these quadrants can simplify computations.

Reference Angles

Reference angles are a powerful tool when it comes to simplifying trigonometric expressions, especially beyond the first quadrant. A reference angle is the acute angle formed by the terminal side of an angle and the x-axis. It represents the smallest angle between the terminal side of a given angle and the horizontal axis.

• This means an angle like 165° or 195° can be related to a reference angle by considering how far it is from the nearest x-axis.
• Reference angles help determine equivalent angles in other quadrants with similar trigonometric values.

For instance, the angle 165° has a reference angle of 15°, which is 180° - 165°. This indicates that trigonometric values at these positions will only differ in sign depending on the quadrant.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable angles. Among these identities, the sum and difference formulas are crucial for computing the trigonometric values of compound angles.

• The cosine of a sum or difference of angles can be determined using: \(\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \).
• This is particularly used to break down complex angles into known values—simplifying calculations without needing a calculator for every angle.

In solving \(\cos(165°)\), first simplifying using the sum formula produces results using known angles like 135° and 30°, aligning the computation with recognizable trigonometric values.