Problem 17 - College Physics

Question

\(\cdot\) An angle is given, to one significant figure, as \(4^{\circ},\) meaning that its value is between \(3.5^{\circ}\) and \(4.5^{\circ} .\) Find the corresponding range of possible values of \((a)\) the cosine of the angle, (b) the sine of the angle, and (c) the tangent of the angle.

Step-by-Step Solution

Answer

Cosine: [0.996, 0.998]; Sine: [0.061, 0.078]; Tangent: [0.061, 0.079].

Step 1: Understand the Range of the Angle

The angle is given as \(4^{\circ}\) and is specified to be between \(3.5^{\circ}\) and \(4.5^{\circ}\). Therefore, our task is to find the cosine, sine, and tangent for the range of angles from \(3.5^{\circ}\) to \(4.5^{\circ}\).

Step 2: Calculate the Cosine Values

To find the cosine values for the angle range, compute the cosine for both the minimum and maximum angles: \(\cos(3.5^{\circ})\) and \(\cos(4.5^{\circ})\). Use a calculator to find these precise values:\[\cos(3.5^{\circ}) \approx 0.998 \quad \text{and} \quad \cos(4.5^{\circ}) \approx 0.996\]Thus, the range of cosine values is approximately \([0.996, 0.998]\).

Step 3: Calculate the Sine Values

Next, find the sine for the angle range. Calculate \(\sin(3.5^{\circ})\) and \(\sin(4.5^{\circ})\):\[\sin(3.5^{\circ}) \approx 0.061 \quad \text{and} \quad \sin(4.5^{\circ}) \approx 0.078\]So, the range of sine values is approximately \([0.061, 0.078]\).

Step 4: Calculate the Tangent Values

Finally, compute the tangent for both the lower and upper bounds:\[\tan(3.5^{\circ}) \approx 0.061 \quad \text{and} \quad \tan(4.5^{\circ}) \approx 0.079\]Thus, the range of tangent values is approximately \([0.061, 0.079]\).

Key Concepts

Understanding Cosine

The cosine of an angle in a right triangle relates the angle to the ratio of the adjacent side over the hypotenuse.
Think of it simply as a way to measure the "horizontal stretch."
When we talk about the `cosine` of small angles, like those near 4 degrees, the values are close to 1 because the triangle is almost like a flat line.

• For our specific angle range between 3.5° and 4.5°, the cosine doesn't vary much.
• As calculated, \[\cos(3.5^{\circ}) \approx 0.998\] li> and \[\cos(4.5^{\circ}) \approx 0.996\].

Thus, our range is narrow, highlighting how cosine is always close to 1 for tiny angles.
Cosine is a fundamental concept because it's often used to resolve forces in physics and calculations in engineering.

Understanding Sine

The sine function measures the "vertical stretch" or the ratio of the opposite side to the hypotenuse in a right triangle.
Similar to cosine, for small angles, sine values are also tiny, but they increase slowly with the angle.

• In our case, \[\sin(3.5^{\circ}) \approx 0.061\]
• and \[\sin(4.5^{\circ}) \approx 0.078\].

The small range between these values shows that even minimal angle changes can slightly increase sine values.
Sine is crucial in scenarios where we need to calculate heights or when dealing with oscillatory motions such as waves.

Understanding Tangent

The tangent of an angle provides the ratio of the sine and cosine, or the opposite side divided by the adjacent side in a triangle.
The `tan` function tells us how steep a line is from the viewpoint of the angle.

• For very small angles like 3.5° to 4.5°, tangent values are also minimal.
• We calculated \[\tan(3.5^{\circ}) \approx 0.061\]
• and \[\tan(4.5^{\circ}) \approx 0.079\].

Here, both values indicate a gentle slope, as expected from these small angles.
`Tangent` has practical applications in physics for calculating speed or angles of elevation and depression in navigation.