Trigonometry for Any Angle. SineCosineTangentCosecantSecantCotangent Abbreviation Reciprocal...

Trigonometry for Any Angle

opposite

adjacent

adjacent

opposite

adjacent

hypotenuse

hypotenuse

adjacent

opposite

hypotenuse

hypotenuse

opposite

cottan

seccos

cscsin

Sine Cosine Tangent Cosecant Secant CotangentAbbreviation

Reciprocal FunctionCo-function

Right Triangle Definition

Unit Circle Definition

Any Angle DefinitionPositive QuadrantsNegative QuadrantsOdd or Even

DomainRangePeriodInverse Inverse Domain

Sine Cosine Tangent Cosecant Secant CotangentAbbreviation sin cos tan csc sec cot

Reciprocal Function

csc sec cot sin cos tan

Co-function cos sin cot sec csc tan

Right Triangle DefinitionUnit Circle Definition



Reciprocal Function




Opp/hyp Adj/hyp Opp/adj Hyp/opp Hyp/adj Adj/opp


y x y/x 1/ y 1/x x/y



Reciprocal Function







Any Angle DefinitionPositive Quadrants

1 and 2 1 and 4 1 and 3 1 and 2 1 and 4 1 and 3

Negative Quadrants


Odd or Even

7

An even function:f(x) = f(-x)

cos(30o) = cos(-30o)?

cos(135o) = cos(-135o)?

The cosine and its reciprocal are evenfunctions.

8

An odd function:f(-x) = -f(x)

sin(-30o) = -sin(30o)?

sin(-135o) = -sin(135o)?

The sine and its reciprocal are oddfunctions.

9

An odd function:f(-x) = -f(x)

tan(-30o) = -tan(30o)?

tan(-135o) = -tan(135o)?

The tangent and its reciprocal are oddfunctions.

10

Cosine and secant functions are evencos (-t) = cos t sec (-t) = sec t

Sine, cosecant, tangent and cotangent are oddsin (-t) = - sin t csc (-t) = - csc ttan (-t) = - tan t cot (-t) = - cot t

Add these to your worksheet

How can we memorize it?SymmetryFor Radians

thedenominatorshelp!

Knowing the quadrant givesthe correct+ / - sign

ON YOUR OWN try these… Write the question and the answer

3cos.1

13

6sin.2

14

6

5sin.3

15

3

2cos.4

16

2cos.5

17

3sin.6

18

6

13cos.7

19

2

3sin.8

20

)(sin.9

21

4

3tan.10

22

3cos.1

2

1

24

6sin.2

2

1

25

6

5sin.3

2

1

26

3

2cos.4

2

1

27

2cos.5

0

28

3sin.6

2

3

29

6

13cos.7

2

3

30

2

3sin.8

1

31

)sin(.9

0

32

4

3tan.10

1

33

Let θ be an angle in standard position. Its reference angle is the acute angle θ’ (called “theta prime”) formed by the terminal side of θ and the horizontal axis.

Let θ be an angle in standard position and its reference angle has the same absolute value for the functions, the sign ( +/ - ) must be determined by the quadrant of the angle.

Quadrant II θ’ = π – θ (radians)

= 180o – θ (degrees) Quadrant III θ’ = θ – π (radians)

= θ – 180o (degrees) Quadrant IV θ’ = 2π – θ (radians)

= 360o – θ (degrees)

Given a point on the terminal side Let be an angle in standard position with

(x, y) a point on the terminal side of and r be the length of the segment from the origin to the point

022 yxr

r

θ

(x,y)

Then….

The six trigonometric functions can be defined as

0,cot0,sec0,csc

0,tancossin

yy

xxx

ryy

r

xx

y

r

x

r

y

Add these definitions to summary worksheet


Reciprocal Function







Any Angle Definition

y/r x/r y/x r/y r/x x/y

Positive Quadrants


Negative Quadrants


Odd or Even Odd Even Odd Odd Even Odd

Find sin, cos and tan given (-3, 4) is a point on the terminal side of an angle. 1. Find r2. Find the ratio of the sides of the reference

angle3. Make sure you have the correct sign based

upon quadrantFind r. (-3)2 + (4)2= r2

r =5

sin θ = 4/5

cos θ = -3/5

tan θ = -4/3

θ r

(-3, 4)

The cosine and sine of the angle are positive1The cosine and sine are negative3The cosine is positive and the sine is negative.4The sine is positive and the tangent is negative2The tangent is positive and the cosine is negative.3The secant is positive and the sine is negative.4

Given tan = -5/4 and the cos > 0, find the sin and sec .

Which quadrant is it in?

The tangent is negative, and the cosine is positive

Quadrant IV at point (4, -5)

θ

r(4, -5)

Find r and use the triangle to find the sine and secant

Let be an angle in quadrant II such that sin = 1/3 find the cos and the tan .

θ 3

(x, 1)

Set up a triangle based upon the information given .

Calculate the other side

Find the other trigonometric functions

Trigonometry for Any Angle. SineCosineTangentCosecantSecantCotangent Abbreviation Reciprocal...

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