Question 1204697: Use the information below to find tan(a+b).
cosa=8/17 with a in quadrant 1.
tanb=5/12 with b in quadrant 3.
Provide the exact answer, not a decimal approximation.
Found 2 solutions by Theo, math_tutor2020:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you are given:
cosa=8/17 with a in quadrant 1.
tanb=5/12 with b in quadrant 3.
from that, you can derive:
tana = 15/8 in quadrant 1.
tanb = -5/-12 = 5/12 in quadrant 3.
to get the exact answer, you need the tan(a+b0 formula.
that is:
tan(a+b) = (tana + tanb) / (1 - tana * tanb)
that becomes:
tan(a+b) = (15/8 + 5/12) / (1 - 15/8 * 5/12)
do the math and you get:
tan(a+b) = 220/21
i confirmed that answer is correct, by doing the following:
220/21 = 10.47619048 in my calculator.
angle a in the first quadrant = arccos(8/17) = 61.92751306 degrees.
angle b in the third quadrant = arctan(5/12) = 202.6198649 degrees.
angle (a + b) = the sum of those two = 264.547378 degrees.
tan(a + b) = tan(64.547378) = 10.47619048.
i got tan(a + b) = 10.47619848 both ways, confirming that the answer is correct.
try it yourself to see if you get the same answer.
let me know how you did.
theo
the trigonomeric identities are fairly easy to find on the web.
for this one, i just put in "tan(a + b) in goodle search and got the identity and lot of information about it at https://www.cuemath.com/trigonometry/tan-a-plus-b/
i also did a search on trigonmetric identities and, one of the websites that came up was https://www2.clarku.edu/faculty/djoyce/trig/identities.html.
thre is no shortage of information on the web.
sometimes you have to sift through it to get what you want.
most of the time, it can be found.
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
Answer: tan(a+b) = 220/21
Explanation
Trig formulas to memorize (or have on a reference sheet).
| sin(theta) = y/rcos(theta) = x/rtan(theta) = y/x | csc(theta) = r/ysec(theta) = r/xcot(theta) = x/y |
Technically you only need to memorize 3 formulas, since each column is the reciprocal of the other.
Example: sine is the reciprocal of cosecant.
(x,y) is the location of the terminal point
r = sqrt(x^2+y^2) = distance from origin to terminal point
r = sqrt(x^2+y^2) = distance from (0,0) to (x,y)
x = adjacent
y = opposite
r = hypotenuse
Angle 'a' is in quadrant 1 where 0 < a < 90.
Any terminal point in this quadrant has x > 0 and y > 0.
cos(a) = 8/17 leads to x = 8 = adjacent and r = 17 = hypotenuse
Then use the pythagorean theorem to solve  and you should find that y = 15
Therefore cos(a) = 8/17 leads to sin(a) = 15/17 when angle 'a' is in quadrant 1.
Note how we have a 8-15-17 right triangle. It's one of the infinitely many pythagorean triples.
Because angle 'a' is in quadrant 1, sin(a) > 0.
Angle b is in quadrant 3 where 180 < b < 270.
Both x and y are negative here.
tan(b) = 5/12 means y = -5 and x = -12
Plug those into and you'll find that r = 13
We have a 5-12-13 right triangle.
So,
cos(b) = x/r = -12/13
sin(b) = y/r = -5/13
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To summarize so far
sin(a) = 15/17
cos(a) = 8/17
sin(b) = -5/13
cos(b) = -12/13
Then we'll use a trig identity
sin(a+b) = sin(a)*cos(b)+cos(a)*sin(b)
sin(a+b) = (15/17)*(-12/13)+(8/17)*(-5/13)
sin(a+b) = -220/221
Use another trig identity
cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b)
cos(a+b) = (8/17)*(-12/13) - (15/17)*(-5/13)
cos(a+b) = -21/221
Divide the results to find the tangent
tan(a+b) = sin(a+b)/cos(a+b)
tan(a+b) = sin(a+b) divide cos(a+b)
tan(a+b) = (-220/221) divide (-21/221)
tan(a+b) = (-220/221) * (-221/21)
tan(a+b) = 220/21
Another way to reach this is to use the trig identity
tan(a+b) = (tan(a) + tan(b))/(1 - tan(a)*tan(b))
that tutor @Theo had used.
GeoGebra can be used to verify the answer.
More practice with trigonometry
https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1204615.html
and
https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1204374.html
and
https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1204248.html
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