Finding exact values for the tangent of the sum or difference of two angles is a little more complicated, but again, it is a matter of recognizing the pattern.
Finding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and simplifying. Recall,
Let’s derive the sum formula for tangent.
We can derive the difference formula for tangent in a similar way.
Sum and difference formulas for tangent
The
sum and difference formulas for tangent are:
Given two angles, find the tangent of the sum of the angles.
Write the sum formula for tangent.
Substitute the given angles into the formula.
Simplify.
Finding the exact value of an expression involving tangent
Find the exact value of
Let’s first write the sum formula for tangent and then substitute the given angles into the formula.
Next, we determine the individual function values within the formula:
We can use the sum and difference formulas to identify the sum or difference of angles when the ratio of sine, cosine, or tangent is provided for each of the individual angles. To do so, we construct what is called a reference triangle to help find each component of the sum and difference formulas.
To find
we begin with
and
The side opposite
has length 3, the hypotenuse has length 5, and
is in the first quadrant. See
[link] . Using the Pythagorean Theorem, we can find the length of side
Since
and
the side adjacent to
is
the hypotenuse is 13, and
is in the third quadrant. See
[link] . Again, using the Pythagorean Theorem, we have
Since
is in the third quadrant,
The next step is finding the cosine of
and the sine of
The cosine of
is the adjacent side over the hypotenuse. We can find it from the triangle in
[link] :
We can also find the sine of
from the triangle in
[link] , as opposite side over the hypotenuse:
Now we are ready to evaluate
We can find
in a similar manner. We substitute the values according to the formula.
For
if
and
then
If
and
then
Then,
To find
we have the values we need. We can substitute them in and evaluate.
Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
Rafael
vjuvu
Elgoog
what is big bang theory?
Rosemary
what type of activity astronomer do?
Rosemary
No
Richard
the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
according to the theory of astronomers why the moon is always appear in an elliptical orbit?
Gatjuol
hi !!! I am new in astronomy....
I have so many questions in mind ....
all of scientists of the word they just give opinion only.
but they never think true or false ...
i respect all of them...
I believes whole universe depending
on true ...থিউরি
Govinda
hello
Jackson
hi
Elyana
we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe
there many theory to born universe but what is the reality of big bang theory to born universe
Asmit
what is the exact value of π?
Nagalakshmi
by big bang
universal
there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
Aarya
I think after Big Bang!
Michele
from where on earth could u observe all the stars during the during the course of an year
is that so. the question was in the end of this chapter
Karuna
in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*).
So it would just barely creep over the horizon.
Christopher
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