Problem 38 - Algebra and Trigonometry

Question

In Exercises \(33-42,\) let $$ \sin t=a, \cos t=b, \text { and } \tan t=c $$ Write each expression in terms of \(a, b,\) and \(c .\) $$ \sin (t+2 \pi)+\cos (t+4 \pi)-\tan (t+\pi) $$

Step-by-Step Solution

Answer

The equation \(\sin(t+2 \pi)+\cos(t+4 \pi)-\tan(t+\pi)\) written in terms of \(a, b, c\) is \(a + b - c\).

Step 1: Express \(\sin(t+2\pi)\) in terms of \(a\)

Notice that the sine function has a period of 2\(\pi\). Therefore, adding 2\(\pi\) to its argument doesn't change its value. Hence, \(\sin(t + 2\pi) = \sin(t) = a\).

Step 2: Express \(\cos(t+4\pi)\) in terms of \(b\)

Similarly, the cosine function also has a period of 2\(\pi\). Therefore, adding 4\(\pi\) to its argument doesn't change its value. Hence, \(\cos(t + 4\pi) = \cos(t) = b\).

Step 3: Express \(\tan(t+\pi)\) in terms of \(c\)

On the other hand, the tangent function has a period of \(\pi\). Therefore, adding \(\pi\) to its argument changes the sign of its value. Hence, \(\tan(t + \pi) = -\tan(t) = -c\).

Step 4: Substitute into expression

Substitute the values of \(\sin(t + 2\pi)\), \(\cos(t + 4\pi)\), and \(\tan(t + \pi)\) from Steps 1, 2, and 3 into the expression to get it in terms of \(a, b, c\). \(\sin(t + 2\pi) + \cos(t + 4\pi) - \tan(t + \pi) = a + b - c\).

Key Concepts

Sine Function

The sine function, often denoted as \( \sin \theta \), is one of the primary functions in trigonometry. Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. The sine of an angle in a right triangle is the ratio of the length of the side opposite to the angle to the length of the hypotenuse.
\[ \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \]
This function is periodic, which means it repeats its values in regular intervals. Specifically, it has a period of \( 2\pi \) radians, or 360 degrees. So, if you add or subtract any multiple of \( 2\pi \) to its argument, you will end up with the same sine value:

• \( \sin(t + 2\pi) = \sin(t) \)

This periodicity is very useful when simplifying trigonometric expressions.

Cosine Function

The cosine function, represented as \( \cos \theta \), is another fundamental trigonometric function. The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle:
\[ \cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} \]
Much like the sine function, the cosine function is periodic with a period of \( 2\pi \). This periodicity means that the function's value repeats every \( 2\pi \) radians, which is why:

• \( \cos(t + 4\pi) = \cos(t) \)

Understanding this property allows us to simplify expressions involving the cosine function by reducing them to their fundamental form, regardless of large additions or subtractions of \( \pi \) or \( 2\pi \).

Tangent Function

The tangent function is another key function in trigonometry and is defined as the ratio of the sine and cosine functions:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
This function has a different periodicity than sine and cosine. The tangent function repeats its values every \( \pi \) radians, meaning it has a period of \( \pi \). When the tangent function is shifted by \( \pi \) radians, the sign of its value changes:

• \( \tan(t + \pi) = -\tan(t) \)

This property is crucial for simplifying expressions that involve the tangent function. It allows for adjustments in trigonometric equations by accounting for the change in sign when incrementing or decrementing by \( \pi \) radians. Remembering this can be very helpful when dealing with problems that involve shifting the angle by integer multiples of \( \pi \).