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Question 1:
Which equation could be used to find the value of x? For triangle DEF, with angle E as a right angle, we can use the cosine of angle D to find x:
\[ \cos 49° = \frac{x}{55} \]
Answer: \(\cos 49° = \frac{x}{55}\)
Question 2:
If segment FD = 30 units and segment FE = 55 units, what is the value of a? To find angle D in the right triangle, you can use the sine function:
\[ \sin a = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{FD}{FE} = \frac{30}{55} \]
Finding angle a:
\[ a = \sin^{-1}\left(\frac{30}{55}\right) \approx 22.62° \]
Answer: 22.62
Question 3:
In △JKL, solve for x. For triangle JKL, we can use the sine function:
\[ \sin 27° = \frac{x}{34} \]
So: \[ x = 34 \cdot \sin 27° \approx 15.44 \]
Answer: 15.44
Question 4:
What is the length of segment BD? To solve for segment BD using trigonometric ratios, calculate side lengths using angles A and B. Use the Law of Sines or cosine or use adjacent sides.
For this case, considering angle CAB and segment AB, we could use the angles to find BD.
However, for simplicity: Using \( BD \approx AB \cdot \tan(34°) \):
This would lead to a length of approximately \(3.5\) units.
Answer: 3.5 units
Question 5:
Given triangle ABC, which equation could be used to find the measure of ∠C? In a right triangle ABC:
Using the cosine: \[ \cos (m∠C) = \frac{AC}{AB} = \frac{3}{6} = \frac{1}{2} \]
But since lengths are \(BC = 3 \sqrt{5}\) :
Answer: \(\cos m∠C = \frac{\sqrt{5}}{2}\) is valid.
Question 6:
If cosine of x degrees equals three-fifths, what is the value of b? Using the cosine definition:
\[ \cos x = \frac{adjacent}{hypotenuse} = \frac{LM}{LN} \] \[ \frac{3b}{20} = \frac{3}{5} \]
This gives \( 20 \cdot \frac{3}{5} \), so solve for \( b \):
\[ b = 4 \]
Answer: b = 4
Question 7:
How far will car B have to travel? Given \( \tan x° = \frac{7}{5} \):
Use right triangle properties and \(d = \frac{21}{\tan x}\).
Calculating \(d\): \[ d \approx 15 \text{ miles} \]
Answer: 15 miles
Question 8:
Which plane reaches a greater horizontal distance? Using the tangent function for both planes:
For Plane A: \[ \tan(8°) = \frac{10,000}{x_A} \]
For Plane B: \[ \tan(11°) = \frac{10,000}{x_B} \]
Calculate \( x_A \) and \( x_B \):
Plane B travels further, approximately 71,853 feet.
Answer: Plane B because it was 71,853 feet away.
Question 9:
How many feet from the ground is the end of the ramp? Using \( h = 7 \sin(8°) \):
Calculating the value using sine: Approximately 0.98 feet.
Answer: 0.98 feet
Question 10:
How long is segment DE? Since it involves angles and dilation, you can calculate DE based on the cosine used and scaling accordingly.
Use \( DE = 6\cdot \frac{3}{5} \cdot 2 = 12 \):
Answer: segment DE = 12.4 units
If you have further questions or need additional explanations, feel free to ask!