Find the Other Trig Values in Quadrant I cos(theta)=4/5

Problem

cos(θ)=4/5,θ∈Quadrant I

Solution

1. Identify the given values for the right triangle. Since cos(θ)=adj/hyp we have the adjacent side x=4 and the hypotenuse r=5

2. Apply the Pythagorean theorem to find the opposite side y In Quadrant I, all trigonometric values are positive, so y will be positive.

x2+y2=r2

4+y2=5

16+y2=25

y2=9

y=3

3. Calculate the sine value using the ratio sin(θ)=opp/hyp

sin(θ)=3/5

4. Calculate the tangent value using the ratio tan(θ)=opp/adj

tan(θ)=3/4

5. Determine the reciprocal identities for cosecant, secant, and cotangent.

csc(θ)=1/sin(θ)=5/3

sec(θ)=1/cos(θ)=5/4

cot(θ)=1/tan(θ)=4/3

Final Answer

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

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