the length of the side labeled x is approximately 41.4 units. The answer is not among the provided options A. 57.9, B. 69.4, C. 61.1, or D. 36.9.
find the length of the side labeled **x** in the given right triangle. We have the following information:
- Angle: 39°
- Opposite side: 26
1. First, we'll find the value of sin(39°):
- Using trigonometry, we know that **sin(39°)** is the ratio of the opposite side to the hypotenuse:
[tex]\[ \sin(39°) = \frac{{\text{{opposite side}}}}{{\text{{hypotenuse}}}} \][/tex]
- We can express the hypotenuse using the Pythagorean theorem:
[tex]\[ \text{{hypotenuse}} = \sqrt{{\text{{opposite side}}^2 + x^2}} \][/tex]
- Rearranging the equation:
[tex]\[ \sin(39°) = \frac{{26}}{{\sqrt{{26^2 + x^2}}}} \][/tex]
2. Next, we'll solve for x:
- Cross-multiplying:
[tex]\[ 26\sqrt{{26^2 + x^2}} = 26 \][/tex]
- Squaring both sides:
[tex]\[ 26^2(26^2 + x^2) = 26^2 \][/tex]
- Simplifying:
[tex]\[ 26^2x^2 = 0 \][/tex]
- Solving for x:
[tex]\[ x = \frac{{26}}{{\sin(39°)}} \][/tex]
3. Calculating the value of x:
- Using the value of sin(39°)(approximately 0.6293):
[tex]\[ x \approx \frac{{26}}{{0.6293}} \approx 41.4 \][/tex]
Therefore, the length of the side labeled x is approximately 41.4 units. The answer is not among the provided options A. 57.9, B. 69.4, C. 61.1, or D. 36.9.
