Question

a point on the terminal side of angle theta is given. Find the exact value of each of the six trigonometric functions of theta. given point- (-12, 5)

Answer 1

a point on the terminal side of angle theta is given. Find the exact value of each of the six trigonometric functions of theta.



given point- (-12, 5)

see the attached figure to better understand the problem

we have that

the point P lies on the II quadrant

that means

the function cosine is negative

the function sine is positive

the function tangent and cotangent are negative

the function secant is negative and the function cosecant is positive

step 1

cos(x)

Applying the Pythagorean Theorem find out the hypotenuse of the right triangle

c^2=12^2+5^2

c^2=144+25

c^2=169

c=13

cos(x)=12/13

Remember that the cosine of angle theta is negative

therefore

`\cos (\theta)=-(12)/(13)`

step 2

Find out sin(x)

sin(x)=5/13

`\sin (\theta)=(5)/(13)`

step 3

Find out tan(x)

tan(x)=5/12

`\tan (\theta)=-(5)/(12)`

step 4

cot(x)

cot(x)=12/5

`\cot (\theta)=-(12)/(5)`

step 5

sec(x)

sec(x)=1/cos(x)

sec(x)=13/12

`\sec (\theta)=-(13)/(12)`

step 6

csc(x)

csc(x)=1/sin(x)

csc(x)=13/5

`\csc (\theta)=(13)/(5)`