I II III IV Sin & csc are + Sin & csc are + Cos & sec are + Tan & cot are + Sin & csc are + Cos & sec are - Tan & cot are - I II III IV Sin & csc are - Cos & sec are + Tan & cot are - Sin & csc are - Cos & sec are - Tan & cot are +

Name the quadrant that ø lies in based on signs. Sin ø > 0 and cos ø < 0 II Tan ø < 0 and Sec ø < 0 Csc ø < 0 and cot ø > 0 III

sin = y/r csc = r/y cos = x/r sec = r/x tan = y/x cot = x/y Here are the trig identities that we already know and will continue to use. sin = y/r csc = r/y cos = x/r sec = r/x tan = y/x cot = x/y

X r Y cos2 θ + sin2 θ = 1 This is the first pythagorean identity. Pythagorean identities involving trig functions. We know from Geometry that the pythagorean theorem is a2 + b2 = c2 If we apply this to the unit circle as we have seen: x2 + y2 = r2 because the x and y are the sides of a right triangle of any point on the unit circle and because the unit circle has a radius of 1 then x2 + y2 = 1. θ Using the trig identities that sin θ = y and cos θ = x we can substitute in the trig functions into the pythagorean theorem and get cos2 θ + sin2 θ = 1 This is the first pythagorean identity. Keep in mind that θ is referring to any angle on the unit circle; Ex. Θ = π/3, π/4, π/6, π/2

Everything up to now has been on a unit circle (radius = 1) Lets try it with an actual point. Example: Find the exact value of the other trig functions if sin θ = √3/4 cos θ = √13/4 4 √3 θ √13 point. P(3,-4) What is true about the hypotenuse of the triangle and the radius of the circle? Therefore we can find the trig ratios of any point from the origin (reference angle) just by its x and y. Sin ø = √3/4 Csc ø = 4/√3 Cos ø = √13/4 Sec ø = 4/√13 Tan ø = √3/√13 Cot ø = √13/√3 Using Pythagorean theorem we find that the radius is 4.

Example: #35 Find the exact value of the other trig functions if sin θ = 4/5 cos θ = -3/5 -3 θ 4 5 point. P(3,-4) What is true about the hypotenuse of the triangle and the radius of the circle? Therefore we can find the trig ratios of any point from the origin (reference angle) just by its x and y. Sin ø = 4/5 Csc ø = 5/4 Cos ø =- 3/5 Sec ø = -5/3 Tan ø = -4/3 Cot ø = -4/3 Using Pythagorean theorem we find that the radius is 5.