Ch. 4 Trigonometry

4.1 / 4.2 Radian/Degree Measure and Unit Circle

Essential ?: How do you describe angles and angular movement?How to you evaluate the trig fncs. using a unit circle.

A. Obj: to find the radian/degree measure of an angle; to evaluate the 6 trig fncs. using the unit circle.

B. Facts/Formulas:

1. As derived from the Greek language, the word trigonometry means “measurement of triangles.”

2. An angle is determined by rotating a ray (half-line) about its endpoint.

3. The starting position of the ray is the initial side of the angle, and the position after rotation is the terminal side

4. The endpoint of the ray is the vertex of the angle

5. An angle is in standard position when the vertex is the origin and the initial side corresponds to the x-axis.

6. Counterclockwise rotation generates positive angles and clockwise rotation generates negative angles

7. angles and have the same initial and terminal sides. Such angles are coterminal. Two angles are coterminal when they have the same initial and terminal sides. For instance, the angles 0 and 2 are coterminal, as are the angles / 6 and 13 / 6. You can find an angle that is coterminal to a given angle by adding or subtracting 2 (one revolution),

8

9. Because the measure of an angle of one full revolution is s/r = 2 r/r = 2 radians, you can obtain the following.

10. Angles between 0 and / 2 are acute angles and angles between / 2 and are obtuse angles.

11. Two positive angles and are complementary (complements of each other) when their sum is / 2. Two positive angles are supplementary (supplements of each other) when their sum is .

12. Degree measure - 2 radians corresponds to one complete revolution, degrees and radians are related by the equations

360 = 2 rad and 180 = rad.

13.

14. unit circle given by x2 + y2 = 1

15.

16. Similarly, the unit circle is divided into 12 equal arcs, corresponding to t-values of

4.3/4.4 Right Triangle and Any Angle Trig

Essential ?: How do you use trig to find unknown sides and angles? How do you evaluate trig fncs of any angle?

A. Obj: to use fundamental trig identities; to evaluate trig fncs of any angle; to find reference angles

B. Facts/Formulas:

1.

2. Special Angles:

3. Cofunctions of complementary angles are equal.

sin(90 – ) = cos cos(90 – ) = sin

tan(90 – ) = cot cot(90 – ) = tan

sec(90 – ) = csc csc(90 – ) = sec

4. Identities:

5. Angle of elevation, which represents the angle from the horizontal upward to an object. Angle of depression, which represents the angle from the horizontal downward to an object.

6.

7. Signs of Trig Fncs.

8. Reference Angle

9. Evaluate using Reference <

10. Common Angles:

4.5 Graphs of Sine, Cosine and Tangent

Essential ?: How do you sketch a graph of sine, tangent and cosine?

A. Obj: to find the amplitude and period of the sine, tangent and cosine graph

B. Facts/Formulas:

1. Sine graph:

2. Cosine graph:

3. Domain of the sine and cosine functions is the set of all real numbers. Range of each function is the interval [–1, 1], and each function has a period of 2.

4. Key pts:

5. Amplitute/Period:

y = d + a sin(bx – c) and y = d + a cos(bx – c).

The constant factor a in y = a sin x acts as a scaling factor—a vertical stretch or vertical shrink of the basic sine curve. When | a | > 1, the basic sine curve is stretched, and when | a | < 1, the basic sine curve is shrunk.

*** Note that when 0 < b < 1, the period of y = a sin bx is greater than 2 and represents a horizontal stretching and when b > 1, the period of y = a sin bx is less than 2 and represents a horizontal shrinking

6. Translations:

a. y = a sin(bx – c) and y = a cos(bx – c) creates a horizontal translation (shift) of the basic sine and cosine curves.

b. y = d + a sin(bx – c) and d + a cos(bx – c). The shift is d units up for d > 0 and d units down for d < 0.

4.7 Inverse Trig Fncs

Essential ?: How do you evaluate and graph the inverse of a trig fnc.

A. Obj: to evaluate the graph of an inverse trig fnc.; to evaluate the composition of trig fncs.

B. Facts/Formulas:

1. Inverse Sine Fnc:

y = arcsin x or y = sin –1 x.

2. The notation sin –1 x is consistent with the inverse function notation f –1(x).

3. The arcsin x notation (read as “the arcsine of x”) comes from the association of a central angle with its intercepted arc length on a unit circle. So, arcsin x means the angle (or arc) whose sine is x.

4. Inverse Trig Fnc:

5. Graphs of Inverse:

Domain: (-∞, ∞)Range: (-π2 ,

π2¿

Domain: [–1,1]Range: [0, ]

Domain: [–1,1]Range: