Problem 56 - Algebra 2 and Trigonometry

Question

In \(51-58,\) find the smallest positive value of \(\theta\) to the nearest minute. $$ \cos \theta=0.6534 $$

Step-by-Step Solution

Answer

\(\theta \approx 49^\circ 7'\)

Step 1: Understanding the Problem

We need to find the angle \(\theta\) whose cosine is 0.6534. We are also asked to find the smallest positive value of \(\theta\) and approximate it to the nearest minute.

Step 2: Applying the Inverse Cosine Function

To find \(\theta\), we apply the inverse cosine function: \(\theta = \cos^{-1}(0.6534)\). This calculation will give us \(\theta\) in radians by default on most scientific calculators.

Step 3: Conversion from Radians to Degrees

Once \(\theta\) is obtained in radians, convert it to degrees since we need the answer in degrees and minutes. Use the formula: \(\theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi}\).

Step 4: Extract Degrees and Convert Fractional Part to Minutes

After converting to degrees, split the result into whole degrees and the fractional part. Multiply the fractional part by 60 to convert it to minutes.

Step 5: Rounding and Conclusion

Round the number of minutes to the nearest whole number to get the final answer for \(\theta\) expressed in degrees and minutes.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are special functions that reverse the operation of regular trigonometric functions like sine, cosine, and tangent. They are essential when you need to find the angle corresponding to a given trigonometric value. For example, if you know that the cosine of an angle is 0.6534, the inverse cosine function, written as \(\cos^{-1}(0.6534)\), will give you the angle whose cosine is 0.6534. This process is vital in trigonometry when solving equations and dealing with real-world problems such as physics or engineering.Inverse functions can sometimes be confusing because they have restrictions to maintain their functionality. For instance, the range for the inverse cosine function is between 0 and \(\pi\) radians (or 0 to 180 degrees). This is because cosine is a periodic function, and the inverse needs to map each value uniquely back to one angle within its principal range.

Cosine Function

The cosine function is a fundamental function in trigonometry, often abbreviated as "cos." It relates an angle in a right triangle to the ratio of the length of the adjacent side over the hypotenuse. Mathematically, it is expressed as:

• \( \cos(\theta) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \)

This function is periodic, which means it repeats its values in regular intervals. In the unit circle, the cosine of an angle gives the x-coordinate of the point corresponding to that angle. The function is even, which means \( \cos(\theta) = \cos(-\theta) \).Understanding the cosine function is important for identifying patterns in oscillatory phenomena such as sound waves and alternating current. By solving problems involving the cosine function, like finding a missing angle when the cosine value is known, you delve into the core aspects of trigonometry.

Angle Conversion

Often in mathematics and applied sciences, it is necessary to convert angles from one unit of measure to another, particularly when working between radians and degrees. The two systems are both valid but are used in different contexts:

• Radians are used primarily in advanced mathematics, physics, and engineering because they simplify certain mathematical expressions, particularly those involving calculus.
• Degrees are more common for everyday uses in fields such as geography and navigation.

Converting between these units uses the relationship that \( \pi \) radians is equivalent to 180 degrees:\[\text{Degrees} = \frac{180}{\pi} \times \text{Radians}\]This formula not only makes the conversion simple but also allows you to switch back and forth as needed. Accurate conversion is important for comparing angles accurately across different calculations.

Radians to Degrees

In trigonometry and other branches of mathematics, working with radians can make calculations easier, especially when dealing with periodic functions. However, degrees are often more intuitive for us, which makes conversion essential.To convert an angle from radians to degrees, use the formula:\[\theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi}\]This conversion factor, \(\frac{180}{\pi}\), comes from the definition that a full circle is both 360 degrees and \(2\pi\) radians.For example, consider you calculate an angle as 1 radian. Using the conversion formula, multiply 1 by \( \frac{180}{\pi} \) to get approximately 57.3 degrees.Understanding this conversion is critical, whether you are doing theoretical mathematics or need the practical application in fields such as architecture or flight navigation, where precise angle measurements in degrees are required.