The minimum value of 4 cosθ + 3 is
-1
Understanding Trigonometric Expressions and Minimum Values
The question asks for the minimum value of the expression $4 \cos\theta + 3$. To find the minimum value of such an expression, we need to understand the range of the trigonometric function involved, which in this case is the cosine function, $\cos\theta$.
Range of the Cosine Function (cosθ)
The cosine function, $\cos\theta$, is a periodic function that oscillates between -1 and 1, regardless of the angle $\theta$ (as long as $\theta$ is a real number). This means:
- The maximum value of $\cos\theta$ is 1.
- The minimum value of $\cos\theta$ is -1.
The range of $\cos\theta$ can be written mathematically as $-1 \le \cos\theta \le 1$.
Calculating the Minimum Value of 4 cosθ + 3
We want to find the minimum value of the expression $4 \cos\theta + 3$. Since the coefficient of $\cos\theta$ (which is 4) is positive, the minimum value of the expression will occur when $\cos\theta$ is at its minimum value.
The minimum value of $\cos\theta$ is -1.
Substitute the minimum value of $\cos\theta = -1$ into the expression:
Minimum Value $= 4 \times (\text{minimum value of } \cos\theta) + 3$
Minimum Value $= 4 \times (-1) + 3$
Now, perform the calculation:
Minimum Value $= -4 + 3$
Minimum Value $= -1$
Therefore, the minimum value of the expression $4 \cos\theta + 3$ is -1.
| Component | Range/Value | Effect on Expression |
|---|---|---|
| $\cos\theta$ | $[-1, 1]$ | Determines the value of $4 \cos\theta$ |
| $4 \cos\theta$ | $[4 \times -1, 4 \times 1] = [-4, 4]$ | Range based on $\cos\theta$ range |
| $4 \cos\theta + 3$ | $[-4 + 3, 4 + 3] = [-1, 7]$ | Range of the final expression |
From the table, we can see that the range of the expression $4 \cos\theta + 3$ is $[-1, 7]$. The smallest value in this range is -1, which is the minimum value.
Conclusion: Minimum Value Calculation
By using the known range of the cosine function, we determined that the minimum value of $\cos\theta$ is -1. Substituting this into the given expression $4 \cos\theta + 3$ yields $4(-1) + 3 = -4 + 3 = -1$. Hence, the minimum value is -1.
Revision Table: Key Trigonometry Concepts
| Trigonometric Function | Range | Period |
|---|---|---|
| $\sin\theta$ | $[-1, 1]$ | $2\pi$ or $360^\circ$ |
| $\cos\theta$ | $[-1, 1]$ | $2\pi$ or $360^\circ$ |
| $\tan\theta$ | $(-\infty, \infty)$ | $\pi$ or $180^\circ$ |
| $\csc\theta$ | $(-\infty, -1] \cup [1, \infty)$ | $2\pi$ or $360^\circ$ |
| $\sec\theta$ | $(-\infty, -1] \cup [1, \infty)$ | $2\pi$ or $360^\circ$ |
| $\cot\theta$ | $(-\infty, \infty)$ | $\pi$ or $180^\circ$ |
Additional Information: Finding Max/Min of Trigonometric Expressions
For a general trigonometric expression of the form $A \sin(Bx+C) + D$ or $A \cos(Bx+C) + D$, where $A, B, C, D$ are constants:
- The range of $\sin(Bx+C)$ or $\cos(Bx+C)$ is $[-1, 1]$.
- The range of $A \sin(Bx+C)$ or $A \cos(Bx+C)$ is $[-|A|, |A|]$.
- The maximum value of the expression is $|A| + D$.
- The minimum value of the expression is $-|A| + D$.
In our question, the expression is $4 \cos\theta + 3$. Here, $A=4$ and $D=3$.
- Maximum value $= |4| + 3 = 4 + 3 = 7$.
- Minimum value $= -|4| + 3 = -4 + 3 = -1$.
This confirms our calculation. Understanding the general form helps quickly determine the range and hence the maximum and minimum values of such trigonometric expressions.
Important Questions from Trigonometric Functions
If A = cos2θ + sin4θ then for all values of θ is :
If \(\frac{{\sin x + \cos x}}{{\sin x - \cos x}} = \frac{6}{5}\) , then the value of \(\frac{{{{\tan }^2}x + 1}}{{{{\tan }^2}x - 1}}\) is:
What is the value of \(? = \frac{{ta{n^2}{{60}^0} - 2si{n^2}{{45}^0}}}{{cos{{24}^0}cos{{37}^0}coses{{53}^0}cos{{60}^0}cosec{{66}^0} + si{n^2}{{60}^0}}}\)
If Y = tan35°, then the value of (2tan55° + cot55°) is :
If -sin θ + cosec θ = 6, then what is the value of sin θ + cosec θ?
