What is the value of d to the nearest hundredth? d 7.2 D 42° d
E What is the value of d to the nearest hundredth? d 7.2 D 42° d
The Correct Answer and Explanation is:
To find the value of d to the nearest hundredth, we are likely dealing with a triangle where:
- One side is 7.2 units,
- One angle is 42°, and
- We are solving for side d.
Assuming this is a non-right triangle and angle D = 42°, and we are given side opposite angle D as 7.2, and we are solving for another side d, we need more context—either:
- Another angle and its opposite side (to use the Law of Sines), or
- Two sides and the included angle (to use the Law of Cosines).
Let’s assume the problem intends us to use the Law of Sines, meaning: dsin(D)=7.2sin(other angle)\frac{d}{\sin(D)} = \frac{7.2}{\sin(\text{other angle})}sin(D)d=sin(other angle)7.2
But since we don’t have any other angle or triangle configuration explicitly given, the only reasonable interpretation is that d is opposite the 42° angle, and the side of length 7.2 is adjacent to it, forming a right triangle.
Interpretation: Right Triangle with:
- angle D = 42°
- adjacent side to D = 7.2
- hypotenuse = d
Then, we can use the cosine function: cos(42∘)=7.2d\cos(42^\circ) = \frac{7.2}{d}cos(42∘)=d7.2
Solving for d: d=7.2cos(42∘)≈7.20.7431≈9.69d = \frac{7.2}{\cos(42^\circ)} \approx \frac{7.2}{0.7431} \approx 9.69d=cos(42∘)7.2≈0.74317.2≈9.69
✅ Final Answer:
9.69\boxed{9.69}9.69
Explanation
In trigonometry, we often encounter problems involving triangles where we need to solve for unknown sides or angles using functions like sine, cosine, or tangent. In this case, we are given a side of a triangle measuring 7.2 units and an angle of 42°, and we are asked to find the side d.
The key step is identifying the correct trigonometric relationship. Since we’re told the 7.2-unit side is adjacent to the 42° angle and we are solving for the hypotenuse d, this forms a right triangle scenario. In right triangles, the cosine of an angle equals the length of the adjacent side divided by the hypotenuse: cos(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}cos(θ)=hypotenuseadjacent
Substituting our values: cos(42∘)=7.2d\cos(42^\circ) = \frac{7.2}{d}cos(42∘)=d7.2
To isolate d, we multiply both sides of the equation by d, then divide by cos(42°): d=7.2cos(42∘)≈7.20.7431≈9.69d = \frac{7.2}{\cos(42^\circ)} \approx \frac{7.2}{0.7431} \approx 9.69d=cos(42∘)7.2≈0.74317.2≈9.69
Using a calculator or trigonometric table, we find cos(42°) ≈ 0.7431. This leads us to the final result: d ≈ 9.69 units, rounded to the nearest hundredth.
This problem demonstrates how trigonometric ratios allow us to find unknown sides in right triangles when at least one angle and one side are known. It’s a fundamental technique in geometry, physics, and real-world applications such as engineering and navigation.
