Now, we will return to the problem posed at the beginning of the section. A bicycle ramp is constructed for high-level competition with an angle of
formed by the ramp and the ground. Another ramp is to be constructed half as steep for novice competition. If
for higher-level competition, what is the measurement of the angle for novice competition?
Since the angle for novice competition measures half the steepness of the angle for the high level competition, and
for high competition, we can find
from the right triangle and the Pythagorean theorem so that we can use the half-angle identities. See
[link] .
We see that
We can use the half-angle formula for tangent:
Since
is in the first quadrant, so is
Thus,
We can take the inverse tangent to find the angle:
So the angle of the ramp for novice competition is
Double-angle identities are derived from the sum formulas of the fundamental trigonometric functions: sine, cosine, and tangent. See
[link] ,
[link] ,
[link] , and
[link] .
Reduction formulas are especially useful in calculus, as they allow us to reduce the power of the trigonometric term. See
[link] and
[link] .
Half-angle formulas allow us to find the value of trigonometric functions involving half-angles, whether the original angle is known or not. See
[link] ,
[link] , and
[link] .
Section exercises
Verbal
Explain how to determine the reduction identities from the double-angle identity
Use the Pythagorean identities and isolate the squared term.
We can determine the half-angle formula for
by dividing the formula for
by
Explain how to determine two formulas for
that do not involve any square roots.
multiplying the top and bottom by
and
respectively.
For the half-angle formula given in the previous exercise for
explain why dividing by 0 is not a concern. (Hint: examine the values of
necessary for the denominator to be 0.)
Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ...
Step 2: Find each score's deviation from the mean. ...
Step 3: Square each deviation from the mean. ...
Step 4: Find the sum of squares. ...
Step 5: Divide the sum of squares by n – 1 or N.
The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.
Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400
a. what is the probability of getting more than 12,000 hits?
b. what is the probability of getting fewer than 9,000 hits?
Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400.
a. What is the probability of getting more than 12,000 hits