In triangle abc, a=8, b=6 and angle c = 60 degrees. find the measure of angle a.

A ≈ 73.9°

Step-by-step explanation:

You might be expected to use the Law of Cosines to find side c, then the Law of Sines to find angle A.

A shorter method suggests itself when considering the altitude to side AC (see the diagram). The length of that will be 8·sin(60°). The distance from C to the point where the altitude intersects AC is then 8·cos(60°). The tangent of angle A is the ratio of the altitude to the remainder of AC, which will be

... 6 - 8·cos(60°)

That is ...

... tan(A) = 8·sin(60°)/(6 -8·cos(60°))

If we divide numerator and denominator by 8, we have ...

... tan(A) = sin(60°)/(0.75 -cos(60°))

A calculator can then make very short work of finding the value of angle A. (See second attachment.)

... A = arctan(sin(60°)/(0.75 -cos(60°))) ≈ 73.898°

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We know cos(60°) = 0.5 and sin(60°) = (√3)/2, so this reduces to

... A = arctan(2√3)

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More about Trig Relationships

Your familiarity with SOH CAH TOA tells you ...

... Sin = Opposite/Hypotenuse

Solving for Opposite gives ...

... Opposite = Hypotenuse×Sin

Likewise,

... Cos = Adjacent/Hypotenuse

so ...

... Adjacent = Hypotenuse×Cos

These two relations are used to find the sides of triangle CBX (where BX is the altitude of ΔABC). Then length AX is the difference between CX and CA.

Now, the tangent is ...

... Tan = Opposite/Adjacent

In our triangle, that is ...

... tan(A) = BX/AX = (BC×sin(60°))/(AC - BC×cos(60°))

6 on both sides and twelve on the top and the bottom

step-by-step explanation: