The diagonal of a rectangle of width 3.8, makes an angle of 32° 15' with each of the longer sides. Find the length of the diagonal, and the length of the rectangle.
Im not sure how to picture/draw the shape, which in term will tell me if it is cos, sin or tan. Also will someone explain to me what the degree and comma sign means? (after 32 and 15) Thanks in advance.
Ok, I know how to do this question, but I just don't get the apostrophe after 15. Supposedly, it means seconds but I don't know what that has to do with this.
Suggestion:
Sketch a rectangle ABCD, with A at the topleft corner. AB as the width of 3.8 and AD as the length.
Draw the diagonal BD.
Now look at the right-angled triangle BCD
angle DBC = 32° 15'
The unit measure of and angle in common notation is the degree. There are 360° in one complete rotation, (a whole pizza)
Each degree is subdivided into 60 minutes and each minute is again divided into 60 seconds.
( I know, I know, the same units we have for time, very confusing)
so 32° 15' = 32° + 15/60° = 32.25°
You must learn the basic definitions of sine, cosine and tangent in terms of sides of a right-angled triangle.
So here
sin 32.25° = DC/BD
sin 32.25 = 3.8/BD
BD sin32.25 = 3.8
BD = 3.8/sin 32.25 <----- use your calculator
BD = 3.8/.533614515 <---- don't round off here
= 7.12124..
= appr 7.1 units
similarly , tan 32.25 = 3.8/BC
BC = ......
your turn
From bobpursley, one of our math tutors:
It is 32.25 degrees.
The 15' is 15 minutes. 60 min per degree.
NO ONE does those anymore. They do decimals.
I like 'em. People use decimals because of calculators. When I see °'" I run to my trig tables and do some interpolation!
To understand the shape and the angle mentioned in the problem, let's visualize it. Draw a rectangle and label its width as 3.8 units. The longer sides of the rectangle are perpendicular to the width. Now, draw a diagonal inside the rectangle connecting the opposite corners.
Now, let's understand the notation used for angles. In this case, the angle is given as "32° 15'".
The degree symbol "°" represents degrees—a unit of angular measurement.
The single quote mark "'" represents minutes. Each degree is further divided into 60 minutes.
Therefore, in this case, 1 degree is equivalent to 60 minutes.
The value "32° 15'" can be read as "32 degrees and 15 minutes."
To find the length of the diagonal and the length of the rectangle, we can use the concept of trigonometry.
From the problem statement, we know that the angle formed by the diagonal and each of the longer sides is 32° 15'. This means that we can consider a right triangle within the rectangle, with the diagonal as the hypotenuse.
Let's use the sine function to solve this problem.
Sine (θ) = Opposite/Hypotenuse
In our case, the angle θ is 32° 15' and the opposite side is the length of the rectangle. The hypotenuse is the length of the diagonal.
Let's denote the length of the diagonal as 'd' and the length of the rectangle as 'l'.
Now, applying sine to our right triangle:
sin(32° 15') = l/d
To find the values of the sine function using a scientific calculator or an online calculator, follow these steps:
1. Convert the angle to decimal form.
- The degrees remain the same: 32°
- For the minutes, divide the value by 60 and add it to the degrees: (15/60) = 0.25
32° + 0.25 = 32.25°
2. Use a calculator to find the sine value of 32.25 degrees: sin(32.25°) ≈ 0.5367
So, the equation becomes:
0.5367 = l/d
To find the length of the rectangle, we need to know the value of 'd'.
Now, let's find the length of the diagonal, 'd'.
To do this, we rearrange the equation:
d = l / 0.5367
We know that the width of the rectangle is given as 3.8 units. We can substitute this information into the equation:
d = 3.8 / 0.5367 ≈ 7.08 units
Therefore, the length of the diagonal is approximately 7.08 units.
Now, we can find the length of the rectangle, 'l', by multiplying the width of the rectangle (3.8 units) by the sine value (0.5367):
l = 3.8 * 0.5367 ≈ 2.04 units
Therefore, the length of the rectangle is approximately 2.04 units.