LECTURE: 1-5: TRIGONOMETRY REVIEW AND INVERSETRIGONOMETRIC FUNCTIONS

Basic Trigonometry

In this class you are expected to have some basic understanding and proficiency in trigonometry. Specifically, youshould know (a) the triangle definitions of all six trigonometric functions, (b) the definitions of the four non-sineand cosine trigonometric functions in terms of sine and cosine, (c) be able to graph all six trigonometric functions,(d) be familiar with the unit circle definition and be able to evaluate all trigonometric functions at common angleswithout the use of a calculator, (e) memorize and be able to use six basic, trigonometric identities.

The Triangle Defintion

Example 1: Sketch a right triangle with side a adjacent to an angle ✓, o opposite of the angle ✓ and hypotenuse h.Define each of the six trigonometric functions in terms of that triangle.

a) sin ✓

b) cos ✓ c) tan ✓ d) sec ✓ e) csc ✓ f) cot ✓

Functions in Terms of Sine and Cosine

Example 2: Define the following four functions in terms of sine and cosine. How does this relate to your answersto Example 1?

(a) tan ✓

(b) sec ✓ (c) csc ✓ (d) cot ✓

The Unit Circle Approach

Example 3: Using the 45-45-90 triangle and 30-60-90 triangle find the coordinates on the unit circle in Quadrant 1.What pattern do you see? What coordinate on the unit circle gives sine? What coordinate gives cosine? Does thisagree with the triangle definition of sine and cosine? Now, reflect the points in quadrant 1 into quadrant 2.

Day 5 1 1-5 Trigonometry

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Example 4: Using the unit circle idea, find sine and cosine of the following angles. Don’t look back at page 1.Rather, sketch a small unit circle and ask yourself these questions (1) what quadrant?, (2) what triangle or axis?,(3) are you looking for x (horizontal) or y (vertical)?

(a) sin(� 2⇡3 )

(b) cos(

11⇡4 ) (c) cos(

3⇡2 )

Example 5: Find the following values.

(a) tan(

3⇡4 )

(b) cot(

⇡

6 ) (c) sec(⇡)

Example 6: Graph the following functions on [�⇡,⇡]. Clearly label any asymptotes.

a) f(x) = tanx

x

y

b) f(x) = cotx

x

y

Example 7: Graph the following functions on [�2⇡, 2⇡]. Clearly label any asymptotes.

a) f(x) = secx

x

y

b) f(x) = cscx

x

y

Day 5 2 1-5 Trigonometry

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Common Identities

When you first learned trigonometry you probably had to prove a bunch of equivalencies using various identities.Perhaps you discussed why these identies were true. Some of you may have been asked to memorize all of theseand some of you may have been allowed a note card/ cheat sheet. In this class you need to have the followingSIX identities and no more.The Pythagorean Identities: Looking at the unit circle, derive an identity involving sine and cosine using thepythagorean theorem. This is (a). Next, divide the answer to (a) by cos

2✓ to obtain a new identity for (b). Finally,

divide (a) by sin

2✓ to obtain the third Pythagorean identity for (c).

(a) Identity #1

(b) Identity #2 (c) Identity #3

The Half/ Double Angle Identities: You have probably seen the identities cos(2✓) = 1 � 2 sin

2✓ and cos(2✓) =

1 + 2 cos

2✓. It is more likely that we will use these slightly differently in this class. Solve these identities for sin2 ✓

and cos

2✓ respectively. Finally, given an identity for sin(2✓).

(a) cos(2✓) = 1� 2 sin

2✓

(b) cos(2✓) = 1 + 2 cos

2✓ (c) sin(2✓)

Inverse Trigonometric Functions

Definition and graphs of sin�1x and cos

�1x

Example 9: Evaluate the following expressions.

(a) sin

�1(

p3/2)

(b) cos

�1(�1/2) (c) tan(arcsin 1/5)

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Definition and graph of tan�1x

Example 10: Simplify the following expressions.

a) tan

�1(�1)

b) tan

�1(

p3) c) tan

�10

Example 11: Simplify the following expressions.

a) sin(tan

�1 x

2 )b) sin(2 arccosx)

Example 12: How is tan

�1(1) different from solving the equation tanx = 1? Why does the former have only one

value while the latter has an infinite number of values?

Example 13: Are the following statements true or false?

(a) sin

�1x =

1sin x

(b) tan

�1(�1) =

3⇡4 ,

7⇡4 (c) sin

2x = 1 + cos

2x

Day 5 4 1-5 Trigonometry

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