Find RT in right triangle RTS, where angle T is 42 degrees and side TS is 6.
Understand the Problem
The question asks us to find the length of side RT in a right-angled triangle RTS, where angle T is 42 degrees and the hypotenuse TS is 6. We can use trigonometric ratios to find the side RT.
Answer
$RT = 4.5$
Answer for screen readers
$RT = 4.5$
Steps to Solve
- Identify the relevant trigonometric ratio
Since we have the angle $T = 42^\circ$ and the hypotenuse $TS = 6$, and we want to find the length of the side $RT$, which is adjacent to angle $T$, we will use the cosine function.
- Set up the equation
The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. Therefore, we have: $$ \cos(T) = \frac{RT}{TS} $$ $$ \cos(42^\circ) = \frac{RT}{6} $$
- Solve for RT
Multiply both sides of the equation by 6: $$ RT = 6 \cdot \cos(42^\circ) $$
- Calculate the value
Using a calculator, we find that $\cos(42^\circ) \approx 0.7431$. Therefore, $$ RT \approx 6 \cdot 0.7431 \approx 4.4586 $$
- Round to the nearest tenth
Rounding $4.4586$ to the nearest tenth gives $4.5$.
$RT = 4.5$
More Information
The cosine function is useful for finding the length of the adjacent side when you know the hypotenuse and the angle.
Tips
A common mistake is using the wrong trigonometric function (e.g., sine or tangent instead of cosine). Another mistake is incorrectly setting up the ratio, for example, using $\frac{6}{RT}$ instead of $\frac{RT}{6}$. Finally, students may forget to round to the nearest tenth as instructed.
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