Chapter 5Section 5.2

Trigonometric Functions

Angles in Standard Position

Recall an angle in standard position is an angle that has its initial side on the positive x-axis. We can use any point on the angles terminal side to find the values of the trigonometric ratios. If the coordinates of the point P are (x,y) and the distance the point P is from the origin is r we get the following values for the trigonometric ratios.

P:(x,y)

x-axis

y-axis

x

y

22 yxr

22sin

yx

yry

22cos

yx

xrx

xy

tan

yx

cot

xyx

xr 22

sec

yyx

yr 22

csc

In the example to the right with the coordinates of P at the point (1,3)

1 2

123

P:(1,3)

10

31 22

r

rr103sin

101cos

13tan

31cot

110sec

310csc

It is important to realize that it does not matter what point you select on the terminal side of the angle the trigonometric ratios will be the same because the triangles are similar. The triangle with its vertex at P1 is similar to the triangle with its vertex at P2 and the length of the sides are proportional (equal ratios).

P1

P2

Signs of Trigonometric Functions

The trigonometric ratios now are defined no matter where the terminal side of the angle is. It can be in any if the four quadrants. Since the values for the xy-coordinates are different signs (±) depending on the quadrant the trigonometric ratios will be also. The value for r is always positive. The chart below shows the signs of the trigonometric ratios.

x pos (+)y pos (+)

x neg (-)y pos (+)

x pos (+)y neg (-)

x neg (-)y neg (-)

Quadrant sin cos tan cot sec csc I + + + + + +II + - - - - +III - - + + - -IV - + - - + -

Find the values of the six trigonometric functions for the left side of the line . The point (-3,2) is on the terminal side of the angle. We find the value for r (distance from the origin) first.

13492)3( 22 r -3

2r

132sin

133cos

32tan

313sec

23cot

213csc

Angle x y sin cos tan cot sec csc

1 0 0 1 0 undefined 1 undefined

0 1 1 0 undefined 0 undefined 1

-1 0 0 -1 0 undefined -1 undefined

0 -1 -1 0 undefined 0 undefined -1

1 0 0 1 0 undefined 1 undefined

0 ,360

90

180

270

1 0.5 0.5 1

1

0.5

0.5

1

Any point on one side of a line can be used to determine the values of the trigonometric functions. To find the values at the 4 angles , , , , and use the 4 points pictured and keep in mind .

Reciprocal Identities:sin 𝜃=

𝑦𝑟 =

1csc𝜃

csc 𝜃=𝑟𝑦=

1sin𝜃

cos𝜃=𝑥𝑟 =

1sec 𝜃

sec𝜃=𝑟𝑥=

1cos𝜃

tan𝜃=𝑦𝑥=

1cot 𝜃

cot 𝜃=𝑥𝑦=

1tan 𝜃

Quotient Identities:

Pythagorean Identities:sin2𝜃+cos2𝜃=1 1+ tan2𝜃=sec2𝜃 1+cot 2𝜃=csc2𝜃These come from the fact that in a right triangle with sides x,y, and r we have:

sin2𝜃+cos2𝜃=𝑥2

𝑟 2+𝑦2

𝑟 2=𝑥2+ 𝑦2

𝑟2=𝑟 2

𝑟 2=1

Trigonometric Identities can be very useful when trying to do the following problem.

If is an angle in the second quadrant and find the other trigonometric functions of t.

611cos

3611cos

1cos3625

1cos65

2

2

22

t

t

t

t

Second quadrant cosine is negative

56

sin1csc

11116

6111

cos1sec

511

tan1cot

11115

115

61165

cossintan

611cos

65sin

tt

tt

tt

ttt

t

t

Besides using Identities you can equate parts of fractions.

If and the value of is in the third quadrant, find the value of the other six trigonometric functions.

In the third quadrant y is negative.