If sinθ = \(\frac{4}{5}\) , Find the value of sin3θ
Finding sin3θ Value from sinθ
The problem asks us to find the value of sin3θ given that sinθ = \( \frac{4}{5} \). To solve this, we need to use the trigonometric triple angle formula for sine.
Understanding the Triple Angle Formula for Sine
The triple angle formula for sin3θ expresses it in terms of sinθ. The formula is:
\[ \sin3\theta = 3\sin\theta - 4\sin^3\theta \]This formula is essential for solving problems involving the sine of a triple angle when the sine of the single angle is known.
Applying the Formula with the Given Value
We are given sinθ = \( \frac{4}{5} \). We need to substitute this value into the formula for sin3θ.
Substitute \( \sin\theta = \frac{4}{5} \) into the formula \( \sin3\theta = 3\sin\theta - 4\sin^3\theta \):
\[ \sin3\theta = 3\left(\frac{4}{5}\right) - 4\left(\frac{4}{5}\right)^3 \]Step-by-Step Calculation
Let's calculate each part of the expression:
- Calculate \( 3\sin\theta \): \[ 3\sin\theta = 3 \times \frac{4}{5} = \frac{12}{5} \]
- Calculate \( \sin^3\theta \): \[ \sin^3\theta = \left(\frac{4}{5}\right)^3 = \frac{4^3}{5^3} = \frac{64}{125} \]
- Calculate \( 4\sin^3\theta \): \[ 4\sin^3\theta = 4 \times \frac{64}{125} = \frac{256}{125} \]
Now, substitute these results back into the formula for sin3θ:
\[ \sin3\theta = \frac{12}{5} - \frac{256}{125} \]To subtract these fractions, we need a common denominator. The least common multiple of 5 and 125 is 125. Convert \( \frac{12}{5} \) to a fraction with a denominator of 125:
\[ \frac{12}{5} = \frac{12 \times 25}{5 \times 25} = \frac{300}{125} \]Now perform the subtraction:
\[ \sin3\theta = \frac{300}{125} - \frac{256}{125} = \frac{300 - 256}{125} = \frac{44}{125} \]Final Value of sin3θ
The calculated value of sin3θ is \( \frac{44}{125} \).
Let's summarize the steps:
| Step | Description | Calculation |
|---|---|---|
| 1 | Given sinθ | \( \sin\theta = \frac{4}{5} \) |
| 2 | Formula for sin3θ | \( \sin3\theta = 3\sin\theta - 4\sin^3\theta \) |
| 3 | Substitute and calculate \( 3\sin\theta \) | \( 3 \times \frac{4}{5} = \frac{12}{5} \) |
| 4 | Substitute and calculate \( \sin^3\theta \) | \( \left(\frac{4}{5}\right)^3 = \frac{64}{125} \) |
| 5 | Substitute and calculate \( 4\sin^3\theta \) | \( 4 \times \frac{64}{125} = \frac{256}{125} \) |
| 6 | Substitute into formula | \( \frac{12}{5} - \frac{256}{125} \) |
| 7 | Find common denominator and subtract | \( \frac{300}{125} - \frac{256}{125} = \frac{44}{125} \) |
Revision Table: Trigonometric Formulas
It's helpful to remember common trigonometric identities and formulas for exam preparation. Here are a few related formulas:
| Formula | Description |
|---|---|
| \( \sin^2\theta + \cos^2\theta = 1 \) | Pythagorean Identity |
| \( \sin2\theta = 2\sin\theta\cos\theta \) | Double Angle Formula for Sine |
| \( \cos2\theta = \cos^2\theta - \sin^2\theta \) \( \cos2\theta = 2\cos^2\theta - 1 \) \( \cos2\theta = 1 - 2\sin^2\theta \) |
Double Angle Formulas for Cosine |
| \( \tan2\theta = \frac{2\tan\theta}{1 - \tan^2\theta} \) | Double Angle Formula for Tangent |
| \( \cos3\theta = 4\cos^3\theta - 3\cos\theta \) | Triple Angle Formula for Cosine |
| \( \tan3\theta = \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta} \) | Triple Angle Formula for Tangent |
Additional Information: Solving Trigonometry Problems
When solving trigonometry problems, especially those involving identities and formulas, consider these tips:
- Identify the Goal: What value or expression are you trying to find?
- Check Given Information: What values or relationships are provided?
- Recall Relevant Formulas: Think about which trigonometric identities or formulas connect the given information to the goal. For this problem, identifying the sin3θ formula was key.
- Substitute Carefully: Plug the given values into the chosen formula correctly.
- Simplify Expressions: Perform algebraic manipulations, combine terms, and simplify fractions step-by-step to avoid errors.
- Use Pythagorean Identity: If sinθ is given, you can find cosθ using \( \sin^2\theta + \cos^2\theta = 1 \), which might be useful in other types of problems. In this specific problem, only sinθ was needed for the sin3θ formula.
Mastering trigonometric formulas and practicing their application is crucial for success in trigonometry.
Important Questions from Trigonometric Ratios and Identities
What is the ratio of the greatest to the smallest value of 2 – 2 sin x – sin 2x, 0 ≤ x ≤ (π/2)?
If x, y are acute angles, where 0 < x + y < 90° and sin(3x - 40°) = cos (3y + 40°), then the value of tan (x + y) is equal to
What is (1 + cot θ - cosec θ)(1 + tan θ + sec θ) equal to?
If \(\sin \theta =\frac{3}{5}\) and \(\cos \theta =\frac{4}{5}\) , then the value of \(\frac{1+\tan \theta}{1-\cot \theta}\) is:
If cos x = p/q and 0° < x < 90°, then the value of tan x is:
