Find the Other Trig Values in Quadrant III tan(theta)=4/3

Problem

tan(θ)=4/3,θ∈Quadrant III

Solution

1. Identify the definitions of the trigonometric functions in terms of x y and r In Quadrant III, both x and y are negative.

2. Determine the values of y and x using the given tangent ratio tan(θ)=y/x Since tan(θ)=4/3 and we are in Quadrant III, we set y=−4 and x=−3

3. Calculate the radius r using the Pythagorean identity r=√(,x2+y2)

r=√(,(−3)2+(−4)2)

r=√(,9+16)

r=√(,25)=5

4. Evaluate the sine and cosine functions using sin(θ)=y/r and cos(θ)=x/r

sin(θ)=(−4)/5

cos(θ)=(−3)/5

5. Evaluate the reciprocal functions csc(θ)=1/sin(θ) sec(θ)=1/cos(θ) and cot(θ)=1/tan(θ)

csc(θ)=−5/4

sec(θ)=−5/3

cot(θ)=3/4

Final Answer

sin(θ)=−4/5,cos(θ)=−3/5,csc(θ)=−5/4,sec(θ)=−5/3,cot(θ)=3/4

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