Question

Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

Answer 1

(a) The value of tan 30 is √3/3.

(b) The value of csc 45 is √2

(c) The value of cot π/3 is√3/3

(d) The value of tan π/4 + csc π/6 is 3.

How to calculate the value of the expressions?

(a) The value of tan 30 is calculated as follows;

tan 30 = 1 / √3

1/√3 = √3/3

tan 30 = √3/3

(b) The value of csc 45 is calculated as follows;

csc 45 = 1 / sin 45

sin 45 = 1/√2

csc 45 = √2

(c) The value of cot π/3 is calculated as follows;

cot π/3 = 1 / tan π/3

π/3 = 180 / 3

π/3 = 60

tan 60 = √3/1

1/tan60 = 1 / √3

1 / √3 = √3/3

cot π/3 = √3/3

(d) The value of tan π/4 + csc π/6

π/4 = 180/4 = 45

π/6 = 180/6 = 30

tan π/4 + csc π/6 = tan 45 + csc 30

tan 45 = 1

csc 30 = 1 / sin30

sin 30 = 1/2

1/sin30 = 2

tan 45 + csc 30 = 1 + 2

tan 45 + csc 30 = 3

Answer 2

we have

`\begin{gathered} tan(30^o)=(1)/(√(3))*(√(3))/(√(3))=(√(3))/(3) \\ therefore \\ tan(30^o)=(√(3))/(3) \end{gathered}`
`\begin{gathered} csc45^o=(1)/(sin45^o) \\ \\ sin45^o=(1)/(√(2)) \\ \\ csc45^o=(1)/((1)/(√(2))) \\ \\ csc45^o=√(2) \end{gathered}`
`\begin{gathered} cot(\pi)/(3)=(1)/(√(3))*(√(3))/(√(3))=(√(3))/(3) \\ \\ cot(\pi)/(3)=(√(3))/(3) \end{gathered}`
`\begin{gathered} tan(\pi)/(4)+csc(\pi)/(6)=(1)/(1)+(1)/((1)/(2))=1+2=3 \\ therefore \\ tan(\pi)/(4)+csc(\pi)/(6)=3 \end{gathered}`