Trigonometric Functions of Any Angle MATH 109 - Precalculus S. Rook

Overview • Section 4.4 in the textbook: – Trigonometric functions of any angle – Reference angles – Trigonometric functions of real numbers 2

Trigonometric Functions of any Angle

Trigonometric Functions of Any Angle • Given an angle θ in

standard position

and a point (x, y) on the terminal side of θ, then

the six trigonometric functions of ANY ANGLE θ

are can be defined in terms of x, y, and the length of the line connecting the origin and (x, y) denoted as r 4

Trigonometric Functions of any Angle (Continued)

Function Abbreviation

The sine of θ The cosine of θ The tangent of θ The cotangent of θ The secant of θ The cosecant of θ sin θ cos θ tan θ cot θ sec θ csc θ

Definition

y r x r y

,

x x x

,

y y

 0  0

r x r

,

x

,

y y

  0 0

r



x

signs from (x, y) 2 

y

5

Trigonometric Functions of any Angle (Continued)

Function Abbreviation

The sine of θ The cosine of θ The tangent of θ The cotangent of θ The secant of θ The cosecant of θ sin θ cos θ tan θ cot θ sec θ csc θ

Definition

y r x r y

,

x x x

,

y y

 0  0

r x r

,

x

,

y y

  0 0

r



x

signs from (x, y) 2 

y

6

Algebraic Signs of Trigonometric Functions • The sign of the six trigonometric functions depends on which quadrant θ terminates in:

r

is the

distance ALWAYS

positive from the origin to (x, y) so it is – The signs of x and y depend on which quadrant (x, y) lies – Remember the shorthand notation involving “the element of” symbol: •  terminates in Q IV 7

Algebraic Signs of Trigonometric Functions (Continued)

Functions θ Є QI θ Є QII θ Є QIII θ Є QIV

sin  

r y

and csc  

r y

+ + – – cos  

x r

tan  

y x

and sec  

r x

and cot  

x y

+ + – – – + + – 8

Trigonometric Functions of any Angle (Example)

Ex 1:

Find the value of all six trigonometric functions if: a) (-1, 2) lies on the terminal side of θ b) (-7, -1) lies on the terminal side of θ 9

Trigonometric Functions of any Angle (Example)

Ex 2:

Given sec θ = 3 ⁄ 2 where cos θ < 0, find the exact value of tan θ and csc θ 10

Reference Angles

Reference Angles • • An important definition is the reference angle – Allows us to calculate ANY angle θ

equivalent positive acute angle using an

• We can now work in all four quadrants of the Cartesian Plane instead of just Quadrant I!

Reference angle: acute angle

denoted θ’, the

positive

that lies between the terminal side of θ and the

x-axis

θ

MUST

be in standard position 12

Reference Angles Examples – Quadrant I Note that both θ and θ’ are 60° 13

Reference Angles Examples – Quadrant II 14

Reference Angles Examples – Quadrant III 15

Reference Angles Examples – Quadrant IV 16

Reference Angles Summary • Depending in which quadrant θ terminates, we can formulate a general rule for finding reference angles: – For any positive angle θ, 0° ≤ θ ≤ 360°: • If θ Є QI: θ’ = θ • If θ Є QII: θ‘ = 180° – θ • If θ Є QIII: θ‘ = θ – 180° • If θ Є QIV: θ’ = 360° – θ 17

Reference Angles Summary (Continued) – If θ > 360°: • • Keep subtracting 360° from θ until 0° ≤ θ ≤ 360° Go back to the first step on the previous slide – If θ < 0°: • Keep adding 360° to θ until 0° ≤ θ ≤ 360° • Go back to the first step on the previous slide – If θ is in radians: • Either replace 180° with π and 360° with 2π OR • Convert θ to degrees 18

Reference Angles (Example)

Ex 3:

i) draw θ in standard position ii) draw θ’, the reference angle of θ: a) 312° c) 4π ⁄ 5 e) 11π ⁄ 3 b) π ⁄ 8 d) -127° 19

Trigonometric Functions of Real Numbers

Reference Angle Theorem •

Reference Angle Theorem:

trigonometric function of an angle θ is

EQUIVALENT

to the

VALUE

the value of a of the trigonometric function of its reference angle – The ONLY thing that may be different is the sign • Determine the sign based on the trigonometric function and which quadrant θ terminates in – The

Reference Angle Theorem

we need to memorize the exact values of 30°, 45°, and 60°

only in Quadrant I!

is the reason why 21

Evaluating a Trigonometric Function Exactly • To evaluate a trigonometric function of θ: – Ensure that 0 < θ < 2π when using radians or 0° < θ < 360° when using degrees – Find θ’ the reference angle of θ – Evaluate the function using the

EXACT

values of the reference angle and the quadrant in which θ terminates • Write the function in terms of sine or cosine if necessary 22

Evaluating a Trigonometric Function (Exactly)

Ex 4:

Give the exact value: a) sin 225° c) tan 120° b) cos 750° d) sec 11π ⁄ 4 23

Summary • • • After studying these slides, you should be able to: – Calculate the trigonometric function of ANY angle θ – State the reference angle of an angle θ in standard position – Evaluate a trigonometric function using reference angles and exact values Additional Practice – See the list of suggested problems for 4.4

Next lesson – Graphs of Sine & Cosine Functions (Section 4.5) 24