Hi 3eb-ga,
Thanks for your question - the first step when looking at this type of
problem is to draw out what is being described. Since trigonometry is
the calculation of distance and angles of right triangles, we can
always classify each side of the triangle relative to a given angle.
Remember that each right triangle contains two legs and a hypotenuse.
In your example you are given an angle of 64.3 degrees. After drawing
this out you can see that this angle is opposite to the tree.
Therefore the leg of the right triangle represented by the tree is
called the "opposite" side.
Since we know the opposite side, we can then label the "adjacent
side", which is the other leg of the triangle. This is the side of the
triangle represented by the ground (we know the value of this side is
115 feet).
The tangent of an angle is a ratio of (opposite side / adjacent side),
or for short:
tan(x) = opp/adj
tan(64.3 deg) = opp/115
115*tan(64.3) = opp
opp = 238.95 feet
Now in addition to the tangent, there are two other common trig
functions called sine and cosine.
Sine of an angle is the ratio of the opposite side to the hypotenuse.
The formula is:
sin(x) = (opposite side / hypotenuse) = opp/hyp
We would use the sine function when we are given an angle and need to
determine either the opposite side or the hypotenuse (where we are
given the other side in the question).
Cosine of an angle is the ratio of the adjacent side to the
hypotenuse. The formula is:
cos(x) = (adjacent side / hypotenuse) = adj/hyp
We would use the cosine function when we are given an angle and need
to determine either the adjacent side or the hypotenuse (where we are
given the other side in the question).
Here are the steps to following general:
1. Draw out a diagram of the problem using the information in the question
2. Based on the angle that you are given, label the sides of the right
triangle (opposite, adjacent, and hypotenuse). With some practice you
will be able to identify the sides without labelling them.
3. Identify which side you want to calculate, and then select the
correct trig function to use based on the information you've been
provided. So if you have an angle and an adjacent side and need to
calculate the hypotenuse, the only equation that works is cosine.
4. Plug the information you have into the appropriate equation and
solve for the desired variable.
As a rule for this type of question, you will always be given one
angle and the distance of one leg. I believe what confuses most people
is the identification of the different sides. Once you can do that, it
is just a matter of identifying what you have, what you are looking
for, and solving the equation that uses all three pieces of
information.
I hope this has given you a deeper understanding of the basic
trigonometric functions. If you have problems understanding any of the
information above, please post a clarification and I will respond
promptly.
Thanks for using Google Answers!
Cheers,
answerguru-ga |
Clarification of Answer by
answerguru-ga
on
23 Nov 2005 11:18 PST
Hi 3eb-ga,
In response to your clarification I am including details on Cosecant,
Secant, and Cotangent. These were not originally included because they
are simply alternate formulas.
The cotangent of an angle is a ratio of (adjacent side / opposite
side), or for short:
cot(x) = adj/opp
We would use the cotangent function when given an angle and need to
determine either the opposite or adjacent side (where we are given the
other side in the question). This is the same condition as tangent, so
you could use either one.
Using your original tree example, you could easily solve the problem
using cotangent:
tan(64.3) = opp/115
1/tan(64.3) = 115/opp
cot(64.3) = 115/opp
opp = cot(64.3)/115
opp = 238.95 feet
Cosecant of an angle is the ratio of the hypotenuse to the opposite
side. The formula is:
csc(x) = (hypotenuse / opposite side) = hyp/opp
The relationship between sine and cosecant is:
csc(x) = 1/sin(x)
We would use the cosecant function when we are given an angle and need
to determine either the opposite side or the hypotenuse (where we are
given the other side in the question). This is the same condition as
cosine, so you could use either one.
Secant of an angle is the ratio of the hypotenuse to the adjacent
side. The formula is:
sec(x) = (hypotenuse / adjacent side) = hyp/adj
The relationship between cosine and secant is:
sec(x) = 1/cos(x)
We would use the secant function when we are given an angle and need
to determine either the adjacent side or the hypotenuse (where we are
given the other side in the question). This is the same condition as
cosine, so you could use either one.
Cheers,
answerguru-ga
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