1. Calculus, Section 1.4

2. Angle Measurement
Recall that there are two systems of angle measurement, radians and degrees. We will begin by examining radians. I will be approaching the construction of the radian slightly differently than the textbook does, but it will yield equivalent results. In most cases, we will use the letter 𝜃 to denote an angle measure.

3. Angle Measurement
Consider the circle with radius 𝑟 below. 𝜃=1 radian is the measure of the central angle of a circle subtended by an arc length of radius 𝑟.

4. Angle Measurement

5. Angle Measurement
In the previous construction, radian measure increases linearly with respect to arc length. Thus, we can define a multiple of a radian by the measure of a central angle of a circle subtended by the same multiple of the circle’s radius.

6. Angle Measurement
The number 𝜋≈ is a mathematical constant that gives the ratio of the circumference of a circle to its diameter. That is, 𝜋= 𝐶 𝐷 , where 𝐶 is the circumference and 𝐷 is the diameter of a circle. (This will not be proven in this course.) It is obvious that the diameter of a circle is twice the radius. Thus, 𝐷= 2𝑟. Then, the equation above gives 𝜋= 𝐶 2𝑟 , which in turn yields 𝐶=2𝜋𝑟. Thus, a rotation through a full circle has radian measure θ=2𝜋. In general, the rotation through one-𝑛th of a circle has radian measure 𝜃= 2𝜋 𝑛 .

7. Angle Measurement
In general, the rotation through one-𝑛th of a circle has radian measure 𝜃= 2𝜋 𝑛 . A positive rotation (𝜃>0) is a rotation in the counterclockwise direction. A negative rotation (𝜃<0) is a rotation in the clockwise direction.

8. Angle Measurement
We say that two angles are coterminal if they have the same initial and terminal sides. Since a rotation through a full circle has radian measure 𝜃=2𝜋, two angles are coterminal if their measures differ by a multiple of 2𝜋. For example, 𝜋 4 is coterminal with both 9𝜋 4 and − 15𝜋 4 . Your book says that “two radian measures represent the same angle if the corresponding rotations differ by an integer multiple of 2𝜋.” I dissent on technical grounds, but for most intents and purposes, such angles are indistinguishable from each other save in measure.

9. Angle Measurement
Degrees are defined by dividing the circle into 360 equal parts. A degree is of a circle. A rotation through 𝜃 degrees (denoted 𝜃°) is a rotation through the fraction 𝜃/360 of the complete circle.

10. Angle Measurement
As with radian measure, positive degrees indicate rotations in the counterclockwise direction, and negative degrees indicate rotations in the clockwise direction. Since 360° represents a rotation through a full circle, two angles are coterminal if their degree measures differ by a multiple of 360°. For example, 45° is coterminal with both 405° and −675°.

11. Conversion Factors Between Degrees and Radians
Recall that a rotation through a full circle is both 2𝜋 radians and 360°. This produces the equation 2𝜋=360° Dividing both sides by 2, we get 𝜋=180° We can then derive conversion factors between degrees and radians by dividing by one side of the equation or the other. For example, dividing by 𝜋 in the equation above yields 1= 180° 𝜋 , while dividing by 180° yields 1= 𝜋 180° . Thus, to convert from radians to degrees, we should multiply by the conversion factor 180 𝜋 , and to convert from degrees to radians, we should multiply by the conversion factor 𝜋 180 .

12. Example 1
Convert 55° to radians and 0.5 rad to degrees.

13. In-Class Assignment 1.4.1
Convert 120° to radians and 𝜋 24 radians to degrees.

14. Common Angles
These are the most common angles we will be working with in this course. You need to memorize these angles.

15. Sine and Cosine
The trigonometric functions sin 𝜃 and cos 𝜃 can be defined in terms of right triangles. Let 𝜃 be an acute angle in a right triangle, and let us label the sides as in the figure below. Then, sin 𝜃 = 𝑜𝑝𝑝 ℎ𝑦𝑝 = 𝑏 𝑐 and cos 𝜃 = 𝑎𝑑𝑗 ℎ𝑦𝑝 = 𝑎 𝑐 .

16. Sine and Cosine
The preceding definition only makes sense if 𝜃 were between 0 and 𝜋 2 . However, using the unit circle, we can extend this definition to an angle measure of any real number. Beginning at the positive 𝑥-axis, rotate an angle 𝜃 through the unit circle. Observing the terminal side following the rotation, let 𝑃= 𝑥, 𝑦 be the coordinates of the point of intersection between the terminal side and the circle. Define sin 𝜃 as the 𝑦-coordinate of this point and cos 𝜃 as the 𝑥-coordinate of this point.

17. Sine and Cosine
These extended definitions (Sine is the 𝑦-coordinate; cosine is the 𝑥- coordinate.) agree with the right-triangle definitions for sine and cosine. Why? (Hint: Let 0<𝜃< 𝜋 2 .)

18. Parity of Sine and Cosine
As we can see from the figure below, it is obvious that sin −𝜃 =− sin 𝜃 and cos −𝜃 = cos 𝜃 . Thus, sin 𝜃 is an odd function and cos 𝜃 is an even function. (Note, this is an illustration of the parity of sine and cosine, not a rigorous proof.)

19. Sine and Cosine
Although we use a calculator to evaluate sine and cosine for general angles, the standard values listed in Figure 5 and Table 2 appear often and should be memorized. They are derived from a geometric construction of the triangle and the triangle.

20. Sine and Cosine
Although we use a calculator to evaluate sine and cosine for general angles, the standard values listed in Figure 5 and Table 2 appear often and should be memorized. They are derived from a geometric construction of the triangle and the triangle.

21. Graphs of Sine and Cosine
If the angle 𝜃 begins at 𝜃=0 and is rotated around the unit circle in a counterclockwise direction, the 𝑦-coordinate initially rises, peaks at 𝑦=1 when 𝜃= 𝜋 2 , accelerates downward until 𝜃=𝜋, slows down and forms a trough 𝑦=−1 at 𝜃= 3𝜋 2 , and then finally returns to its starting position at 𝜃=2𝜋. In this manner, plotting the 𝑦-coordinate as a function of 𝜃 generates the familiar “sine wave”.

22. Graphs of Sine and Cosine
The “cosine wave” is generated in a similar manner, except the 𝑥- coordinate is used instead of the 𝑦-coordinate. This construction is slightly more difficult to conceptualize, but it yields a curve similar to the one for sine, yet shifted to the left 𝜋 2 units. Please see the next two slides for a GIF visualizing the construction of both the sine and the cosine waves. (Note, the cosine curve is rotated sideways on the first GIF for clarity.)

23. Graphs of Sine and Cosine

24. Graphs of Sine and Cosine

25. sin 𝑥 = sin 𝑥+2𝜋𝑘 cos 𝑥 = cos 𝑥+2𝜋𝑘
Periodic Functions
A function 𝑓(𝑥) is called periodic with period 𝑇 if 𝑓 𝑥+𝑇 =𝑓(𝑥) for all 𝑥 and 𝑇 is the smallest positive number with this property. Thus, the sine and cosine functions are periodic with period 𝑇=2𝜋 because the radian measures 𝑥 and 𝑥+2𝜋𝑘 correspond to the same point on the unit circle for any integer 𝑘.* That is, for any integer 𝑘,
sin 𝑥 = sin 𝑥+2𝜋𝑘 cos 𝑥 = cos 𝑥+2𝜋𝑘
Recall that the unit circle has a circumference of 2𝜋 units, which implies that θ=2𝜋 constitutes a full rotation.

26. Additional Trigonometric Functions
The four additional standard trigonometric functions are defined in terms of sine and cosine: tan 𝑥 = sin 𝑥 cos 𝑥 cot 𝑥 = cos 𝑥 sin 𝑥 sec 𝑥 = 1 cos 𝑥 csc 𝑥 = 1 sin 𝑥 These definitions must be memorized. Notice that cot 𝑥 = 1 tan 𝑥 . Also memorize the right-triangle definitions of sine, cosine and tangent: sin 𝜃 = 𝑜𝑝𝑝 ℎ𝑦𝑝 cos 𝜃 = 𝑎𝑑𝑗 ℎ𝑦𝑝 tan 𝜃 = 𝑜𝑝𝑝 𝑎𝑑𝑗 It is often easier to remember these latter definitions with the acronym SOH-CAH-TOA.

27. Additional Trigonometric Functions
𝑦= tan 𝑥 and 𝑦= cot 𝑥 are periodic with period 𝜋 (Why?), while 𝑦= sec 𝑥 and 𝑦= csc 𝑥 are periodic with period 2𝜋.

28. Computing Values of Trigonometric Functions
Since we know the values of sine and cosine for the angles 0, 𝜋 6 , 𝜋 4 , 𝜋 3 , 𝜋 2 , as well as the fact that sine is odd and cosine is even, it becomes easy to calculate the trigonometric functions of several other angles between 0 and 2𝜋. Example: Find the values of the six trigonometric functions at 𝑥= 4𝜋 3 .

29. In-Class Assignment 1.4.2
Find the values of the six trigonometric functions at 𝑥= 11𝜋 6 .

30. Computing Values of Trigonometric Functions
We can also solve some trigonometric equations without the use of a calculator. Example: Find the angles 𝑥 such that sec 𝑥 =2.

31. Computing Values of Trigonometric Functions
In general, the trigonometric functions of most angles in applied problems cannot be solved by hand. Numerically approximating the values of the trigonometric functions in these situations require the use of reference angles and Maclaurin polynomials. In this course, we will use inverse trigonometric functions, which will be discussed in Section 1.5.

32. Trigonometric Identities
Trigonometric functions satisfy a large number of identities. The most fundamental identity is known as the Pythagorean Identity: sin 2 𝑥 + cos 2 𝑥 =1 Why is this reasonable? Equivalent versions are obtained by dividing the above equation by cos 2 𝑥 and sin 2 𝑥 , respectively: tan 2 𝑥 +1= sec 2 𝑥 1+ cot 2 𝑥 = csc 2 𝑥

33. Trigonometric Identities
The other basic trigonometric identities are given in the box below. These will be used occasionally later in the course, and they are especially useful for converting expressions involving one trigonometric function into an expression involving another trigonometric function.

34. Trigonometric Identities
Example: Find tan 𝜃 if sec 𝜃 = 5 and sin 𝜃 <0.

35. Law of Cosines
We conclude this section with the very important Law of Cosines. This formula is extremely useful for solving certain applied problems, especially in our future discussion of related rates in Section Notice that this law holds for any triangle, not just right triangles.