Find Trig Functions Using Identities

Problem

sin(x)=2/3,cos(x)=√(,5)/3

Solution

1. Identify the given values for the sine and cosine functions.

sin(x)=2/3

cos(x)=√(,5)/3

2. Apply the quotient identity to find the tangent function, where tan(x)=sin(x)/cos(x)

tan(x)=(2/3)/(√(,5)/3)

tan(x)=2/√(,5)

tan(x)=(2√(,5))/5

3. Apply the reciprocal identity for the cosecant function, where csc(x)=1/sin(x)

csc(x)=1/(2/3)

csc(x)=3/2

4. Apply the reciprocal identity for the secant function, where sec(x)=1/cos(x)

sec(x)=1/(√(,5)/3)

sec(x)=3/√(,5)

sec(x)=(3√(,5))/5

5. Apply the reciprocal identity for the cotangent function, where cot(x)=1/tan(x)

cot(x)=√(,5)/2

Final Answer

tan(x)=(2√(,5))/5,csc(x)=3/2,sec(x)=(3√(,5))/5,cot(x)=√(,5)/2

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