What is the angle IAB in the diagram, to one decimal place?
Understand the Problem
The question asks about some property to one decimal place in the diagram shown, and refers to I A I B, this most likely is the angle IAB, so will return that value.
Answer
$31.0^{\circ}$
Answer for screen readers
$31.0^{\circ}$
Steps to Solve
- Identify the relevant right triangle
The problem involves finding the angle $\angle IAB$ (which is assumed to be $\theta$ based on the diagram), where point I is likely the same as point E in the upper left of the full image. We can see a right triangle formed with sides of length 30m and (24m - 6m) = 18m. Point A is where those sides meet. The angle $\theta$ is opposite the side that measures 18m, and adjacent to the side that measures 30m.
- Use the tangent function to relate the angle to the sides
Since we have the opposite and adjacent sides, we use the tangent function: $$ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{18}{30} $$
- Solve for $\theta$ using the inverse tangent function
To find $\theta$, we take the inverse tangent (arctan) of $\frac{18}{30}$: $$ \theta = \arctan\left(\frac{18}{30}\right) $$
- Calculate the value of $\theta$ in degrees
Using a calculator: $$ \theta \approx 30.96375653^{\circ} $$
- Round the result to one decimal place
Rounding to one decimal place, we get: $$ \theta \approx 31.0^{\circ} $$
$31.0^{\circ}$
More Information
The problem involves using trigonometry, specifically the tangent function and its inverse, to find an angle in a right triangle.
Tips
A common mistake is to confuse the opposite and adjacent sides, leading to an incorrect tangent value. Also, not rounding to the specified decimal place is an error. Forgetting to convert radians to degrees (if your calculator is in radian mode) will also lead to a wrong answer.
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