Problem 19 - 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. $$ \cos 75^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \cos 75^{\circ} \) is \( \frac{\sqrt{6} - \sqrt{2}}{4} \).

Step 1: Identify the Breakdown

We can write \(75^{\circ}\) as the sum of \(45^{\circ}\) and \(30^{\circ}\), meaning that \(a = 45^{\circ}\) and \(b = 30^{\circ}\). Substituting into the sum identity for cosine, we have \( \cos 75^{\circ} = \cos(45^{\circ} + 30^{\circ}) \).

Step 2: Apply the Sum Identity for Cosine

Applying the sum identity for cosine, we get \( \cos 75^{\circ} = \cos 45^{\circ} \cos 30^{\circ} - \sin 45^{\circ} \sin 30^{\circ} \).

Step 3: Substitute Known Values

We now substitute known values from the unit circle: \( \cos 45^{\circ} = \frac{\sqrt{2}}{2}, \cos 30^{\circ} = \frac{\sqrt{3}}{2}, \sin 45^{\circ} = \frac{\sqrt{2}}{2}, \sin 30^{\circ} = \frac{1}{2} \). Substituting these values in, we have \( \cos 75^{\circ} = \frac{\sqrt{2}}{2} * \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} * \frac{1}{2} \).

Step 4: Simplify the expression

Simplify the expression to obtain \( \cos 75^{\circ} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} \). Combining like terms, we get \( \cos 75^{\circ} = \frac{\sqrt{6}-\sqrt{2}}{4} \).

Key Concepts

Sum and Difference Identities

Sum and difference identities are essential tools in trigonometry that allow us to simplify expressions or find exact values for trigonometric functions at various angles. In this exercise, we use the cosine sum identity:

• \( \cos(a + b) = \cos a \cdot \cos b - \sin a \cdot \sin b \)

By breaking down the angle \(75^{\circ}\) into \(45^{\circ} + 30^{\circ}\), we can use this identity to calculate \(\cos 75^{\circ}\). This method illustrates how complex angles can be expressed as a combination of simpler, recognizable ones like \(45^{\circ}\) and \(30^{\circ}\). Such identities are also applicable in solving equations and simplifying expressions in various mathematical problems. Always remember, these identities can transform challenging trigonometric questions into manageable calculations with well-known angle values.

Unit Circle

The unit circle is a powerful concept in trigonometry, representing all possible angles and their corresponding sine and cosine values. It is a circle centered at the origin with a radius of one. Here are some key features:

• Coordinates on the circle are defined by \((\cos \theta, \sin \theta)\) for any angle \(\theta\).
• Common angles like \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ},\) and \(90^{\circ}\) have well-known values.

In our example, we used the unit circle to find the cosine and sine values for \(45^{\circ}\) and \(30^{\circ}\). These known values are critical for calculations, making it easier to evaluate trigonometric expressions without a calculator. The unit circle effectively bridges the geometry of circles with trigonometric calculations.

Cosine Function

The cosine function, expressed as \(\cos \theta\), gives the x-coordinate of a point on the unit circle corresponding to an angle \(\theta\). Here are some highlights about the cosine function:

• It is an even function, which means \(\cos(-\theta) = \cos(\theta)\).
• Its range is from -1 to 1, representing the maximum and minimum values on the unit circle.
• It starts at maximum value (1) when \(\theta = 0\) and oscillates smoothly as the angle increases.

In solving \(\cos 75^{\circ}\), the cosine function helps us break down the angle into parts where the function's values can be precisely determined using known identities. The beauty of the cosine function is in its periodic nature and symmetry, which simplifies complex trigonometric problems.

Exact Values

Exact values in trigonometry refer to specific trigonometric functions at common angles that have straightforward rational values or involve square roots, instead of decimal approximations. When solving \(\cos 75^{\circ}\), we rely on these exact values:

• \(\cos 45^{\circ} = \frac{\sqrt{2}}{2}\)
• \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\)
• \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\)
• \(\sin 30^{\circ} = \frac{1}{2}\)

These values are derived from geometric considerations, such as right triangles like the 45-45-90 and 30-60-90 triangles, or from the unit circle. Using these values ensures precision in calculations without the need for calculators, which is especially useful in mathematical proofs and theoretical work.

Angle Addition Formula

The angle addition formula is a key trigonometric identity that allows us to find the trigonometric function of the sum or difference of two angles. This is particularly helpful for determining values of functions at non-standard angles. For cosine, the formula is:

• \( \cos(a + b) = \cos a \cdot \cos b - \sin a \cdot \sin b \)

By applying this formula to \(75^{\circ} = 45^{\circ} + 30^{\circ}\), we derive \(\cos 75^{\circ}\) using known values of \(45^{\circ}\) and \(30^{\circ}\).
The angle addition formula is instrumental for solving equations that involve sums and differences of angles, serving as a foundational tool in both theoretical and applied trigonometry.