Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

Approximate to four decimal places. (a) \(\sin 496.4^{\circ}\) (b) \(\cos 0.65\) (c) \(\tan 105^{\circ} 40^{\prime}\) (d) \(\cot 1030.2^{\circ}\) (e) \(\sec 1.46\) (f) \(\csc 320^{\circ} 50^{\prime}\)

Step-by-Step Solution

Answer

(a) -0.7193 (b) 0.7961 (c) -3.3622 (d) -0.7265 (e) 9.0587 (f) -1.3053

Step 1: Convert Degrees to Radians (a)

To approximate \( \sin 496.4^{\circ} \), we first need to convert the angle from degrees to radians. The formula is: \( \, \text{radians} = \frac{\pi}{180} \times \, \text{degrees} \, \). For 496.4 degrees, this is approximately \( 8.6631 \) radians.

Step 2: Calculate Sine (a)

Now that we have the angle in radians, we can calculate \( \sin 8.6631 \) using a calculator, which gives approximately \( -0.7193 \).

Step 3: Calculate Cosine (b)

The angle is already in radians, so we can directly compute \( \cos 0.65 \) using a calculator, resulting in approximately \( 0.7961 \).

Step 4: Convert Angle and Calculate Tangent (c)

First, convert \( 105^{\circ} 40^{\prime} \) to decimal degrees: \(105^{\circ} + \frac{40}{60} = 105.6667^{\circ} \.\) Then, use a calculator to find \( \tan 105.6667^{\circ} \), which is approximately \( -3.3622 \).

Step 5: Convert Degrees to Radians (d)

Convert \( 1030.2^{\circ} \) to radians: \( \text{radians} = \frac{\pi}{180} \times 1030.2 \), which is approximately \( 17.9813 \) radians.

Step 6: Calculate Cotangent (d)

To find \( \cot 1030.2^{\circ} \), calculate \( \tan 17.9813 \) and then take the reciprocal. \( \tan 17.9813 \approx -1.3768 \), thus \( \cot 1030.2^{\circ} \approx -0.7265 \).

Step 7: Calculate Secant (e)

Since the angle is already in radians, calculate \( \cos 1.46 \) and take the reciprocal to find \( \sec 1.46 \). \( \cos 1.46 \approx 0.1104 \), giving \( \sec 1.46 \approx 9.0587 \).

Step 8: Convert Angle and Calculate Cosecant (f)

Convert \( 320^{\circ} 50^{\prime} \) to decimal degrees: \( 320 + \frac{50}{60} = 320.8333^{\circ} \).Calculate \( \sin 320.8333^{\circ} \approx -0.7661 \), giving \( \csc 320.8333^{\circ} \approx -1.3053 \).

Key Concepts

Radians

To understand trigonometric functions, it's essential to grasp the concept of radians. Radians offer an alternative to degrees for measuring angles, prevalent in advanced mathematics.
When you think of radians, imagine wrapping the radius of a circle around its circumference. One full wrap, or the circumference, is always \(2\pi\) radians.

• Converting between degrees and radians is straightforward. The formula is: \( ext{radians} = \frac{\pi}{180} \times \,\text{degrees}\). This means 180 degrees equals \pi\ radians, making it easy to switch between the two systems.
• This method is crucial, as radians often simplify calculations and equations, particularly in calculus and higher-level mathematics.

Once you convert an angle into radians, you can more effectively compute functions like sine, cosine, and tangent. Remember, most calculators and mathematical software work in radians by default!

Sine

The sine function, often represented as \sin\, is vital in describing periodic phenomena. Whether measuring waves or oscillations, sine is a fundamental concept.
In a right-angled triangle, sine of an angle is the ratio of the opposite side to the hypotenuse.

• The formula is: \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\).
• When dealing with angles in radians, the sine function helps find specific values precisely, useful in trigonometric calculations without approximations.

In the context of the original exercise, after converting degrees to radians, you calculate \(\sin 8.6631\), which yields approximately \(-0.7193\). This process shows how useful and precise the sine function is when working through complex problems.

Cosine

Similar to sine, the cosine function, denoted as \cos\, plays a crucial role in trigonometry and mathematics. Cosine can help us describe rotations and predict behavior in waves.
Defined in a right triangle, it is the ratio of the adjacent side to the hypotenuse.

• The formula is: \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\).
• Its values range from -1 to 1, making it ideal for analyzing circular and periodic patterns.

In exercises involving radians, such as \(\cos 0.65\), it's straightforward: directly calculate the cosine value using a calculator, and you get approximately \0.7961\. This shows how cosine calculations are often simpler when angles are already in radians.

Tangent

The tangent function, represented as \tan\, is another indispensable trigonometric function. Unlike sine and cosine, tangent can grow infinitely large.
In a right triangle, tangent relates the opposite side to the adjacent side.

• The formula is: \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).
• This relationship is crucial for solving problems involving slopes and angles of elevation.

For example, in a problem where you compute \(\tan 105.6667^{\circ}\) after converting from degrees to decimal, you find approximately \-3.3622\. This result exemplifies tangent's utility in handling angles beyond the standard range, easily extending the scope of trigonometric analysis.