Problem 13 - Algebra 2 and Trigonometry

Question

In \(3-22 :\) a. Rewrite each function value in terms of its cofunction. b. Find, to four decimal places, the value of the function value found in a. $$ \cos 130^{\circ} $$

Step-by-Step Solution

Answer

\(\cos 130^{\circ} = -0.6428\) when using its cofunction.

Step 1: Identify Cofunction Identity

The cofunction identity relating cosine and sine is \( \cos(\theta) = \sin(90^{\circ} - \theta) \). We need to rewrite \(\cos 130^{\circ}\) using its cofunction.

Step 2: Apply the Cofunction Identity

Substitute \(130^{\circ}\) into the cofunction identity.\[\cos 130^{\circ} = \sin(90^{\circ} - 130^{\circ})\]This simplifies to:\[\cos 130^{\circ} = \sin(-40^{\circ})\]

Step 3: Simplify the Negative Angle

Since the sine function is odd, \( \sin(-\theta) = -\sin(\theta) \). Therefore,\[\sin(-40^{\circ}) = -\sin(40^{\circ})\]Thus,\[\cos 130^{\circ} = -\sin(40^{\circ})\]

Step 4: Calculate the Function Value

Use a calculator to find \(\sin(40^{\circ})\) and then negate it. Ensure the calculator is set to degree mode. \(\sin(40^{\circ}) \approx 0.6428\) thus:\[-\sin(40^{\circ}) \approx -0.6428\]

Key Concepts

Cosine Function

The cosine function is one of the primary trigonometric functions and is often abbreviated as 'cos'. It evaluates the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. Cosine is pivotal in linking angles to lengths and is defined for all angles, both acute and obtuse.

In mathematical terms:

• For an angle \( \theta \), the cosine is given by: \( \cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \).

The function is periodic with a period of \(360^{\circ}\) or \(2\pi\) radians, meaning it repeats its values every full circle. When exploring related angles, cosine can also be transformed using cofunction identities, which connect it to the sine function in special ways.

Sine Function

The sine function complements the cosine function and is indicated as 'sin'. In a right-angled triangle, it represents the ratio of the length of the opposite side to the hypotenuse.

Mathematically, for an angle \( \theta \):

• \( \sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \)

The sine function has a period of \(360^{\circ}\) or \(2\pi\) radians. It is known to be an "odd function", which means that \( \sin(-\theta) = -\sin(\theta) \). This property allows us to evaluate sine for negative angles easily by taking the negative of the sine of the positive angle. It is crucial for solving equations involving cofunction identities, as seen in the problem where \( \cos(\theta) = \sin(90^{\circ} - \theta) \) is utilized.

Negative Angles

Negative angles rotate in the clockwise direction when measured from the positive horizontal axis in a Cartesian plane. Trigonometric functions handle negative angles having distinct properties:

• The sine of a negative angle, \( \sin(-\theta) \), gives the negative of the sine of the angle, \( -\sin(\theta) \).
• Similarly, for cosine and other trigonometric functions, negative angles reveal symmetrical properties, helping simplify calculations.

Understanding how each trigonometric function reacts to negative inputs is essential. For the sine function, this reflection across the origin is fundamental because it makes the calculations in trigonometry easier and more intuitive especially when involving cofunction identities.

Radian and Degree Conversions

Radian and degree are two units of measuring angles, with degrees being more intuitive and everyday-friendly, and radians being the natural choice in mathematics and physics. Converting between the two is crucial in trigonometry:

• One complete circle is \( 360^{\circ} \) and in radians, it is \( 2\pi \). Thus, \( 180^{\circ} = \pi \text{ radians} \).
• To convert degrees to radians: multiply by \( \frac{\pi}{180} \).
• To convert radians to degrees: multiply by \( \frac{180}{\pi} \).

For example, converting \( 130^{\circ} \) to radians involves computing \( 130^{\circ} \times \frac{\pi}{180} \), resulting in approximately \( 2.2689 \) radians. This conversion allows angles to be represented in a form useful for more advanced applications, especially in calculus and analysis.