If cos(23x+20)°=sin(2x−10)°, find the acute angles of the corresponding right triangle.
A= B=
A skateboarding ramp is 14 inches high and rises at an angle of 19°. How long is the base of the ramp? Round to the nearest inch.
A. 38in
B. 41in
C.43in
D.15in
A pilot flying a helicopter locates a person on the ground. The horizontal distance from the person to the helicopter is 950 feet, and the height of the helicopter above the person is 500 feet. What is the approximate angle of depression from the helicopter to the person, rounded to the nearest whole degree?
A. 58
B.62
C.28
D.32
A buoy is floating in the water near a lighthouse. The height of the lighthouse is 18 meters, and the horizontal distance from the buoy to the base of the lighthouse is 45 meters. What is the approximate angle of elevation from the buoy to the top of the lighthouse, rounded to the nearest whole degree?
A. 22
B. 66
C. 24
D. 68
Two girls are standing 100 feet apart. They both see a beautiful seagull in the air between them. The angles of elevation from the girls to the bird are 20° and 45°, respectively. How high up is the seagull? Round to three decimal places. Show your work.
Hi guys my teacher did some of these problems in class...
For the skateboard one its 41 inches
Pilot is 62 degrees
Buoy is 24 degrees
Hope this helps!
Oobleck that makes no sense brother, its an acute angle how would it be 500/950, there is two separate angles on for each function.
OMG THANK YOU!!!!
Well what if I don't WANT to round?! Ever thought about that?!
To solve these trigonometry problems, we need to use trigonometric identities and formulas. Let's start with the first one.
1. For the equation cos(23x+20)° = sin(2x−10)°, we want to find the acute angles of the corresponding right triangle.
First, we need to convert the given trigonometric functions into their corresponding ratios. Recall that:
- cos(theta) = adjacent/hypotenuse
- sin(theta) = opposite/hypotenuse
So, by converting the given functions, we have:
- cos(23x+20)° = adjacent/hypotenuse
- sin(2x−10)° = opposite/hypotenuse
Since the acute angles of a right triangle are usually represented by A and B, we'll use those variables to represent our unknown angles.
Now, let's equate the corresponding ratios:
cos(23x+20)° = sin(2x−10)°
Using the trigonometric identity: sin(theta) = cos(90° - theta), we can rewrite the right side of the equation:
cos(23x + 20)° = cos(90° - (2x - 10)°)
Now, we can equate the angles inside the cosine functions:
23x + 20 = 90 - (2x - 10)
Simplify the equation:
23x + 20 = 90 - 2x + 10
Combine like terms:
23x + 20 = 100 - 2x
Move all the x-terms to one side:
23x + 2x = 100 - 20
Solve for x:
25x = 80
Divide both sides by 25:
x = 80/25
x = 3.2
Now, we can substitute the value of x back to find the angles A and B:
A = 23x + 20 = 23(3.2) + 20 = 73.6°
B = 2x - 10 = 2(3.2) - 10 = -3.6°
However, since we are looking for the acute angles, we consider only the positive values of B:
B = |2x - 10| = |2(3.2) - 10| = 3.6°
So, the acute angles of the corresponding right triangle are:
A = 73.6°
B = 3.6°
Now, let's move on to the other trigonometry problems.
2. To find the length of the base of the skateboarding ramp, given the height and angle, we can use the trigonometric ratio:
- tan(theta) = opposite/adjacent
In this case, the given height is the opposite side of the triangle, and the base is the adjacent side. The angle is given as 19°.
By substituting the values into the formula, we get the equation:
tan(19°) = 14/base
To find the base, we can rearrange the equation:
base = 14/tan(19°)
Using the calculator, we can evaluate:
base ≈ 40.976
Rounding to the nearest inch, the base of the ramp is approximately 41 inches.
Therefore, the correct option is B: 41in.
3. To find the angle of depression from the helicopter to the person, we can use the trigonometric ratio:
- tan(theta) = opposite/adjacent
In this case, the given horizontal distance is the adjacent side, and the height is the opposite side. The angle we're looking for is the angle of depression.
By substituting the values into the formula, we get the equation:
tan(theta) = 500/950
To find the angle, we can use the inverse tangent function (tan⁻¹) to solve for the angle:
theta = tan⁻¹(500/950)
Using a calculator, we find:
theta ≈ 29.987
Rounding to the nearest whole degree, the approximate angle of depression is 30 degrees.
Therefore, the correct option is D: 32.
4. To find the angle of elevation from the buoy to the top of the lighthouse, we can use the trigonometric ratio:
- tan(theta) = opposite/adjacent
In this case, the given horizontal distance is the adjacent side, and the height is the opposite side. The angle we're looking for is the angle of elevation.
By substituting the values into the formula, we get the equation:
tan(theta) = 18/45
To find the angle, we can use the inverse tangent function (tan⁻¹) to solve for the angle:
theta = tan⁻¹(18/45)
Using a calculator, we find:
theta ≈ 21.801
Rounding to the nearest whole degree, the approximate angle of elevation is 22 degrees.
Therefore, the correct option is A: 22.
5. To find the height of the seagull, we can use the tangent function.
Given the angles of elevation from the two girls are 20° and 45°, we can set up the following equations:
- tan(20°) = height/base
- tan(45°) = height/base
Let's solve for the height using the equation with the angle of 45°:
tan(45°) = height/100
Rearranging the equation:
height = tan(45°) × 100
Using the calculator, we find:
height ≈ 100
Therefore, the seagull's height is approximately 100 feet.
since cosθ = sin(90-θ)
23x+20 = 90 - (2x−10)
x/14 = cos19°
tanθ = 500/950
Now you try the rest, after reviewing your basic trig functions.