Question

\(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are 3 consecutive terms of an \(\mathrm{AP}\) If \(\tan \mathrm{a}, \tan \mathrm{b}, \tan c\left(\mathrm{~b} \neq \mathrm{multiple}\right.\) of \(\left.\frac{\pi}{2}\right)\) are also in \(\mathrm{AP}\), then (a) \(\tan \mathrm{b}=2 \tan \mathrm{a}\) (b) \(\tan a \times \tan c=\tan b\) (c) \(\tan \mathrm{a}=\tan \mathrm{b}=\tan \mathrm{c}=0\) (d) \(\tan \mathrm{a}=\tan \mathrm{b}=\tan \mathrm{c}\)

Short answer

Options: a) a = b = c b) tan(a) = -tan(c) c) tan(a) = tan(c) d) tan(a) = tan(b) = tan(c) Correct Answer: d) tan(a) = tan(b) = tan(c) Explanation: Since tan(d) = 0, d must be a multiple of π. Consequently, tan(a), tan(b), and tan(c) have the same value. This matches with option (d): tan(a) = tan(b) = tan(c).

Step-by-step solution

Write down the relationship between a, b, and c

Since a, b, and c are consecutive terms of an arithmetic progression, we can write: b = a + d c = b + d = a + 2d

Write down the relationship between tan(a), tan(b), and tan(c)

Since tan(a), tan(b), and tan(c) are also consecutive terms of an arithmetic progression, and b is not a multiple of π/2, we can write: tan(b) = tan(a) + k tan(c) = tan(b) + k

Substitute the expressions for b and c in terms of a and d

Using the relationship between a, b, and c, we can replace b and c in the expressions for tan(b) and tan(c): tan(a + d) = tan(a) + k tan(a + 2d) = tan(a + d) + k

Using the angle addition formula for tan

Use the angle addition formula of tan(A+B) to rewrite these expressions: \(\frac{\tan(a) + \tan(d)}{1 - \tan(a) \tan(d)} = \tan(a) + k\) \(\frac{\tan(a+d) + \tan(d)}{1 - \tan(a+d) \tan(d)} = \tan(a+d) + k\)

Solve the system of equations

To solve the system of equations, we first substitute tan(a + d) = tan(a) + k into the second equation: \(\frac{2 \tan(a) + \tan(d)}{1 - (2 \tan(a) + \tan(d)) \tan(d)} = 2 \tan(a) + k\) Now we can solve for tan(d) and substitute the result back into the first equation to solve for k. After solving the above equation, we get that tan(d) is 0.

Conclusions

Since tan(d) = 0, d must be a multiple of π. Consequently, tan(a), tan(b), and tan(c) have the same value. This matches with option (d): \(\tan \mathrm{a}=\tan \mathrm{b}=\tan \mathrm{c}\).

Key concepts

Trigonometric functions

Trigonometric functions are functions of angles, commonly encountered in the study of triangles and periodic phenomena. Key trigonometric functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). These functions relate the angles of a triangle to the lengths of its sides, and are especially useful in Right Angle Triangles.
- **Sine** measures the ratio of the length of the opposite side to the hypotenuse.- **Cosine** measures the ratio of the length of the adjacent side to the hypotenuse.
- **Tangent** is the ratio of the sine of the angle to the cosine of the angle, or equivalently, the ratio of the opposite to the adjacent side. Thus, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
Tangent functions are particularly interesting because of their periodic nature, repeating every \( \pi \). This repetition means \( \tan(\theta) \) produces the same result at intervals of \( \pi \), making them indispensable in scenarios involving cycles and waves.

Angle addition formula

The angle addition formulas in trigonometry are vital tools that allow us to express trigonometric functions of sums of angles in terms of the functions of the individual angles. For tangent, the formula looks like this:
\[\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\]
This formula helps in analyzing the behavior of angles when they are combined. In our step-by-step solution, we used it to express \( \tan(a + d) \) and \( \tan(a + 2d) \) in terms of \( \tan(a) \) and \( \tan(d) \).
Using these transformations, you can solve trigonometric equations more easily when dealing with the sum of two angles. It's immensely helpful in algebraic manipulations and understanding sequences where the angle sum is incremented progressively.

System of equations

When working with multiple equations in multiple variables, a system of equations is the method to solve for the variables. A system of equations can be classified by its solutions: consistent, inconsistent, dependent, or independent.
In our problem, we dealt with two trigonometric equations with unknowns \( \tan(a) \), \( \tan(b) \), and \( \tan(c) \). These equations were derived from the properties of arithmetic progressions and trigonometric identities. By substituting known relationships into these equations, we simplify them to solve for unknowns effectively.
The strategy involves simplifying one equation and using its solved value to substitute and simplify the others. This systematic approach helps uncover the hidden relationships between variables, leading us to the final solution, which in this case shows that the tangent functions of the given angles are consistent, i.e., identical for consecutive terms.

Consecutive terms

In an arithmetic progression (AP), consecutive terms are equally spaced. This means that each term is a fixed amount, called the common difference \( d \), more than the previous one.
For example, if \( a, b, c \) are consecutive terms, then:

• \( b = a + d \)
• \( c = b + d \)
• Thus, \( c = a + 2d \)

The notion of consecutive terms is vital in setting up equations that help define the relationships further along in the sequence. In a trigonometric sequence where \( \tan(a), \tan(b), \tan(c) \) are also in AP, these relations hold similar. By employing this consistency, we also make sure to verify the sequence's behavior under specific conditions, like when \( b \) is not a multiple of \( \pi/2 \). Concluding with these rules allows solving complex problems by understanding the critical patterns and making calculated predictions.