Trig ratio challenge

May 2018
2
0
South Africa
Kindly assist with this trig equation challenge:
Given that √5 tanA=-2 and CosB=8/17 in ∆ABC
State why we may assume that angle C is acute and determine the value of Sin C


Attempt made:
tanA=-2/√5 CosB=8/17
A is obtuse angle of 138° or reflex angle 318.19°
B is an acute angle of61.9° or reflex angle298.1 °.Since it is a right angle,therefore angle C is acute
Not sure the above is correct though!
Second Part: The diagram will be in the 4th quadrant since tan is –ve and cos is +ve
Got stucked here!
 
Feb 2010
739
162
Kindly assist with this trig equation challenge:
Given that √5 tanA=-2 and CosB=8/17 in ∆ABC
State why we may assume that angle C is acute and determine the value of Sin C


Attempt made:
tanA=-2/√5 CosB=8/17
A is obtuse angle of 138° or reflex angle 318.19°
B is an acute angle of61.9° or reflex angle298.1 °.Since it is a right angle,therefore angle C is acute
Not sure the above is correct though!
Second Part: The diagram will be in the 4th quadrant since tan is –ve and cos is +ve
Got stucked here!
What's all this 318.19 degrees and 298.1 degrees stuff? You said "in \(\displaystyle \triangle ABC\)". So all angles are less than 180 degrees. Angle A is obtuse so the other two angles are acute.

\(\displaystyle A+B+C=180^{\circ}\) so \(\displaystyle C=180^{\circ}-(A+B)\). Thus, \(\displaystyle \sin C=\sin(180^{\circ}-(A+B))=\sin(A+B)\).

If \(\displaystyle \tan A = \dfrac{-2}{\sqrt{5}}\) then \(\displaystyle \sin A=\dfrac{2}{3}\) and \(\displaystyle \cos A = \dfrac{-\sqrt{5}}{3}\)

If \(\displaystyle \cos B = \dfrac{8}{17}\) then \(\displaystyle \sin B=\dfrac{15}{17}\).

See if you can finish it.
 
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skipjack

Forum Staff
Dec 2006
21,482
2,471
This seems to be a trick question.

Angle A is negative, and is about -41.81°.

Angle B is about ±61.93°.

Angle C is about ±76.26°.

sin(C) = ±(16 + 15√5)/51.
 
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Feb 2010
739
162
Silly me for thinking that the problem was reasonable and didn't need to be checked.

\(\displaystyle \angle A \approx 138.19^{\circ}\) and \(\displaystyle \angle B \approx 61.93^{\circ}\). So there isn't much room for \(\displaystyle \angle C\) in this triangle.
 
May 2018
2
0
South Africa
Thanks a million @skipjack and Mrtwhs! Your contributions has provided the needed insights to solve this problem. Once again Thank you!