1) Identify the initial amount and the growth rate of the following
1a)y=250(1+0.2)^t
2b) y=9.8(1.35)^t
2) Use special right triangles to state the value of the 6 trig functions for 30 degrees, 45 degrees and 60 degrees.
3) Use the calculator to evaluate the value of the following trig function.
cot 35 degrees 21 minutes 45 seconds
surely you can evaluate the functions at t=0:
1a)y=250(1+0.2)^t at t=0: 250(1.02)^0 = 250
2b) y=9.8(1.35)^t at t=0: 9.8(1.35)^0 = 9.8
For the special triangles, consult your text. These come up very often, so learn them and love them! Just do it. It will save you a lot of time later. Just learn the sin, cos, tan. And recall that the other three functions are their reciprocals.
#3. What, you can't use your calculator?
35 degrees 21 minutes 45 seconds = 35 + 21/60 + 45/3600 = 2829/80°
tan 2829/80° = 0.7096
so, cot = 1/tan = 1.4091
1) For the equation y = 250(1+0.2)^t:
- The initial amount is 250, as it is the value of y when t = 0.
- The growth rate is 0.2, which is the value being added to 1 and then raised to the power of t.
For the equation y = 9.8(1.35)^t:
- The initial amount is 9.8, as it is the value of y when t = 0.
- The growth rate is 1.35, which is the value being multiplied by the initial amount and then raised to the power of t.
2) Using special right triangles, we can state the value of the 6 trigonometric functions for 30 degrees, 45 degrees, and 60 degrees.
For 30 degrees:
- sin(30°) = 1/2
- cos(30°) = √3/2
- tan(30°) = 1/√3
- csc(30°) = 2
- sec(30°) = 2/√3
- cot(30°) = √3
For 45 degrees:
- sin(45°) = √2/2
- cos(45°) = √2/2
- tan(45°) = 1
- csc(45°) = √2
- sec(45°) = √2
- cot(45°) = 1
For 60 degrees:
- sin(60°) = √3/2
- cos(60°) = 1/2
- tan(60°) = √3
- csc(60°) = 2/√3
- sec(60°) = 2
- cot(60°) = 1/√3
3) To evaluate cot(35 degrees 21 minutes 45 seconds), we can use a calculator.
- First, calculate the decimal value of the angle by converting the minutes and seconds to decimal degrees.
- 21 minutes = 0.35 degrees (since there are 60 minutes in a degree, 21/60 = 0.35)
- 45 seconds = 0.0125 degrees (since there are 60 seconds in a minute, 45/3600 = 0.0125)
- Add the decimal values to the given angle: 35 degrees + 0.35 degrees + 0.0125 degrees = 35.3625 degrees.
- Use a calculator to find the cotangent of 35.3625 degrees, which is approximately 0.919790132.
Therefore, cot(35 degrees 21 minutes 45 seconds) ≈ 0.9198.
1) To identify the initial amount and the growth rate of the given exponential functions, we need to analyze the equations.
1a) In the equation y = 250(1 + 0.2)^t, the initial amount is 250, and the growth rate is 0.2.
1b) In the equation y = 9.8(1.35)^t, the initial amount is 9.8, and the growth rate is 1.35.
To identify the initial amount, you is essential to see the base amount or the coefficient of the first term in the equation. In both cases, the base amount is 250 in 1a) and 9.8 in 1b).
To identify the growth rate, you need to observe the exponent next to the base amount in the parentheses. In 1a), the growth rate is 0.2, denoted as 1 + 0.2, and in 1b), the growth rate is 1.35, denoted as 1 + 0.35.
2) To find the values of the six trigonometric functions for 30°, 45°, and 60°, we can use special right triangles to relate the angles with sides.
For 30 degrees:
- Sides of the triangle would be in the ratio 1:√3:2.
- sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3, csc(30°) = 2, sec(30°) = 2/√3, cot(30°) = √3.
For 45 degrees:
- Sides of the triangle would be in the ratio 1:1:√2.
- sin(45°) = √2/2, cos(45°) = √2/2, tan(45°) = 1, csc(45°) = √2, sec(45°) = √2, cot(45°) = 1.
For 60 degrees:
- Sides of the triangle would be in the ratio 1:√3:2.
- sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3, csc(60°) = 2/√3, sec(60°) = 2, cot(60°) = 1/√3.
3) To evaluate the value of the trigonometric function cot 35° 21' 45", you can use a calculator.
- First, convert the angle from degrees, minutes, and seconds (DMS) to decimal degrees.
Degrees: 35
Minutes: 21/60 = 0.35
Seconds: 45/3600 = 0.0125
Decimal degrees = 35 + 0.35 + 0.0125 = 35.3625°
- Once you have the angle in decimal degrees, use the cotangent function in your calculator to find the value.
cot(35.3625°) = 1/tan(35.3625°)
Inputting this into a calculator will give you the numerical value of cot(35° 21' 45").