Problem 102 - Precalculus

Question

Calculate \(\csc 40^{\circ}\) in the following two ways: a. Find \(\sin 40^{\circ}\) to three decimal places and then divide 1 by that number. Write this last result to five decimal places. b. With a calculator in degree mode, enter \(40,\) sin, \(1 / \mathrm{x},\) and round the result to five decimal places.

Step-by-Step Solution

Answer

The calculated value of \( \csc 40^\circ \) is 1.55741.

Step 1: Calculate Sin 40°

Using a calculator, find the sine of 40 degrees. Ensure your calculator is set to degree mode, not radian mode. The sine of 40 degrees is approximately 0.642.

Step 2: Compute the Cosecant Using Sin 40°

The cosecant of an angle is the reciprocal of the sine of that angle. So, calculate the cosecant of 40 degrees by taking the reciprocal of 0.642. \[\csc 40^\circ = \frac{1}{\sin 40^\circ} \approx \frac{1}{0.642} \approx 1.55741\] Round this result to five decimal places, which gives 1.55741.

Step 3: Confirm Cosecant with Calculator Method

Ensure your calculator is still in degree mode. Enter 40, calculate the sine (you should see 0.642), then use the 1/x function to find the reciprocal. The result should be 1.55741 when rounded to five decimal places, confirming the result from Step 2.

Key Concepts

Trigonometric Functions

Trigonometric functions are foundational to understanding relationships within triangles, especially right-angled triangles. They describe how angles and sides interact. Some common trigonometric functions include sine (\( \sin \theta \)), cosine (\( \cos \theta \)), tangent (\( \tan \theta \)), as well as their reciprocal functions cosecant (\( \csc \theta \)), secant (\( \sec \theta \)), and cotangent (\( \cot \theta \)). These functions are crucial for analyzing periodic phenomena and are widely applied in fields like engineering, physics, and astronomy.

• Sine, cosine, and tangent are based on the ratios of two sides of a right triangle relative to an angle.
• Cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent, respectively.

Understanding these functions helps to solve various real-world problems, such as calculating heights, distances, and angles. If you grasp how to calculate and apply these functions, you'll find math much more approachable.

Sine Function

The sine function is a primary trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. When you see \( \sin \theta \), it refers to this ratio. For example, if you're given an angle and need to determine its sine, you essentially calculate how high the opposite side stretches relative to the hypotenuse. In the calculation of \( \csc 40^\circ \), first finding \( \sin 40^\circ = 0.642 \) involves this ratio determination. The sine function is:

• A periodic function, which repeats its values in regular intervals.
• Used to calculate wave behavior, such as tides and sound waves.
• Essential in modeling circular and oscillating phenomena.

In practical scenarios, always ensure your calculator is set to the correct mode to compute sine accurately, as seen when calculating trigonometric properties of angles in degrees.

Degree Mode Calculation

Degree mode calculation is a setting on calculators that ensures angle measurements are interpreted as degrees. It's a crucial distinction since calculators can also be set to radian mode. Degrees are commonly used in navigation, geography, and many classroom settings for simplicity and historical reasons. Degrees divide a circle into 360 parts, with each degree representing a \(\frac{1}{360}\) slice of a circle. This mode is especially pertinent when calculating trigonometric values like \(\sin 40^\circ\) or when finding \(\csc 40^\circ\).When you perform calculations with your calculator:

• Always confirm it's set to degree mode if angles are given in degrees, as miscalculations can occur otherwise.
• Switch between modes based on context—sometimes radians are more appropriate, such as in calculus.

Ensuring correct mode usage is one step closer to accurate and effective mathematics problem solving.