If sin (5x - 25°) = cos(5y + 25°), where 5x - 25° and 5y + 25° are acute angles,then the value of (x + y) is:
18°
Solving the Trigonometric Equation for (x + y)
We are given a trigonometric equation involving sine and cosine of angles that are stated to be acute. The equation is:
\(\sin (5x - 25°) = \cos(5y + 25°)\)
We are also told that the angles \((5x - 25°)\) and \((5y + 25°)\) are acute angles. Acute angles are angles that measure between 0° and 90° (exclusive of 0° and 90°).
Understanding the Key Trigonometric Identity
A fundamental trigonometric identity relates the sine and cosine of complementary angles. Complementary angles are two angles whose sum is 90°. The identity states that:
For any angle \(\theta\), \(\sin \theta = \cos (90° - \theta)\) and \(\cos \theta = \sin (90° - \theta)\).
Alternatively, if \(\sin A = \cos B\), where A and B are acute angles, then \(A + B = 90°\).
Applying the Identity to the Given Equation
Given that \(\sin (5x - 25°) = \cos(5y + 25°)\) and that \((5x - 25°)\) and \((5y + 25°)\) are acute angles, we can apply the identity \(\sin A = \cos B \implies A + B = 90°\).
Let \(A = 5x - 25°\) and \(B = 5y + 25°\).
According to the identity for acute angles:
\(A + B = 90°\)
Substitute the expressions for A and B:
\((5x - 25°) + (5y + 25°) = 90°\)
Solving for the Value of (x + y)
Now, we simplify and solve the equation for \((x + y)\):
\(5x - 25° + 5y + 25° = 90°\)
Combine the terms:
\(5x + 5y + (-25° + 25°) = 90°\)
\(5x + 5y + 0° = 90°\)
\(5x + 5y = 90°\)
Factor out the common factor, 5, from the left side:
\(5(x + y) = 90°\)
To find the value of \((x + y)\), divide both sides of the equation by 5:
\(x + y = \frac{90°}{5}\)
\(x + y = 18°\)
Conclusion
The value of \((x + y)\) is 18°. This is derived directly from the trigonometric identity relating the sine and cosine of complementary acute angles.
| Step | Action | Equation |
|---|---|---|
| 1 | Start with the given equation | \(\sin (5x - 25°) = \cos(5y + 25°)\) |
| 2 | Apply the identity \(\sin A = \cos B \implies A+B = 90°\) for acute angles | \((5x - 25°) + (5y + 25°) = 90°\) |
| 3 | Simplify the equation | \(5x + 5y = 90°\) |
| 4 | Factor out 5 | \(5(x + y) = 90°\) |
| 5 | Solve for \((x + y)\) | \(x + y = 18°\) |
Revision Table: Trigonometric Identities Review
| Identity | Description | Condition |
|---|---|---|
| \(\sin \theta = \cos (90° - \theta)\) | Sine of an angle equals cosine of its complement | For any angle \(\theta\) |
| \(\cos \theta = \sin (90° - \theta)\) | Cosine of an angle equals sine of its complement | For any angle \(\theta\) |
| If \(\sin A = \cos B\), then \(A + B = 90°\) | Sum of angles if sine of one equals cosine of another | Specifically for acute angles A and B (or if A+B is an odd multiple of 90° in general) |
Additional Information: Acute Angles and Complementary Angles
An acute angle is an angle strictly between 0° and 90°. The condition that the given angles are acute is important because the identity \(\sin A = \cos B \implies A + B = 90°\) holds reliably for acute angles. While the identity \(\sin \theta = \cos (90° - \theta)\) is true for all angles, concluding \(A+B=90°\) directly from \(\sin A = \cos B\) requires considering the quadrant of the angles if they are not acute. However, for acute angles, the relationship is straightforward.
Complementary angles are two angles that add up to exactly 90°. The trigonometric identities used in this problem highlight the relationship between the sine and cosine of complementary angles. If one angle is \(\theta\), its complement is \(90° - \theta\).
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