Bicycle ramps made for competition (see
[link] ) must vary in height depending on the skill level of the competitors. For advanced competitors, the angle formed by the ramp and the ground should be
$\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ such that
$\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\theta =\frac{5}{3}.\text{\hspace{0.17em}}$ The angle is divided in half for novices. What is the steepness of the ramp for novices? In this section, we will investigate three additional categories of identities that we can use to answer questions such as this one.
Using double-angle formulas to find exact values
In the previous section, we used addition and subtraction formulas for trigonometric functions. Now, we take another look at those same formulas. The
double-angle formulas are a special case of the sum formulas, where
$\text{\hspace{0.17em}}\alpha =\beta .\text{\hspace{0.17em}}$ Deriving the double-angle formula for sine begins with the sum formula,
Deriving the double-angle for cosine gives us three options. First, starting from the sum formula,
$\text{\hspace{0.17em}}\mathrm{cos}\left(\alpha +\beta \right)=\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta -\mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta ,$ and letting
$\text{\hspace{0.17em}}\alpha =\beta =\theta ,$ we have
Similarly, to derive the double-angle formula for tangent, replacing
$\text{\hspace{0.17em}}\alpha =\beta =\theta \text{\hspace{0.17em}}$ in the sum formula gives
Given the tangent of an angle and the quadrant in which it is located, use the double-angle formulas to find the exact value.
Draw a triangle to reflect the given information.
Determine the correct double-angle formula.
Substitute values into the formula based on the triangle.
Simplify.
Using a double-angle formula to find the exact value involving tangent
Given that
$\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\theta =-\frac{3}{4}\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ is in quadrant II, find the following:
$\mathrm{sin}\left(2\theta \right)$
$\mathrm{cos}\left(2\theta \right)$
$\mathrm{tan}\left(2\theta \right)$
If we draw a triangle to reflect the information given, we can find the values needed to solve the problems on the image. We are given
$\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\theta =-\frac{3}{4},$ such that
$\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ is in quadrant II. The tangent of an angle is equal to the opposite side over the adjacent side, and because
$\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ is in the second quadrant, the adjacent side is on the
x -axis and is negative. Use the
Pythagorean Theorem to find the length of the hypotenuse:
We see that we to need to find
$\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta .\text{\hspace{0.17em}}$ Based on
[link] , we see that the hypotenuse equals 5, so
$\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta =\frac{3}{5},$ and
$\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta =-\frac{4}{5}.\text{\hspace{0.17em}}$ Substitute these values into the equation, and simplify.
In this formula, we need the tangent, which we were given as
$\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\theta =-\frac{3}{4}.\text{\hspace{0.17em}}$ Substitute this value into the equation, and simplify.
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0
then
4x = 2-3
4x = -1
x = -(1÷4) is the answer.
Jacob
4x-2+3
4x=-3+2
4×=-1
4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3
4x=-3+2
4x=-1
4x÷4=-1÷4
x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was 1350 bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after 3 hours?
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?