TRIGONOMETRY
TRIGONOMETRY
CHAPTER 3: RADIAN MEASURE AND THE CIRCULAR FUNCTIONS
1. RADIAN MEASURE
➢ RADIAN:
o The figure below shows an angle [pic] in standard position along with a circle of radius r. The vertex of [pic] is at the center of the circle. Angle [pic] intercepts an arc on the circle equal in length to the radius of the circle. Therefore, angle [pic] is said to have a measure of 1 radian.
r r
[pic]
r
➢ CONVERTING BETWEEN DEGREES AND RADIANS
o The circumference of a circle (the distance around a circle) is given by the formula [pic], where r is the radius of the circle. This tells us that there are [pic] radians around a circle. Consider an angle of 360°. This is a complete circle, so 360° intercepts an arc on the circle equal in length to [pic] times the radius of the circle. Therefore, we have [pic] This gives us the following relationship:
[pic]
So we have,
[pic]
• Example: Convert 45° to radians
[pic]
• Example: Convert [pic] to degrees
[pic]
• Example: Find [pic]
[pic]
➢ EQUIVALENT ANGLE MEASURES IN DEGREES AND RADIANS
[pic] |sin [pic] |cos [pic] |tan [pic] |cot [pic] |sec [pic] |csc [pic] | |0° or 0 radians |0 |1 |0 |Undefined |1 |Undefined | |30° or [pic] radians |[pic] |[pic] |[pic] |[pic] |[pic] |2 | |45° or [pic] radians |[pic] |[pic] |1 |1 |[pic] |[pic] | |60° or [pic] radians |[pic] |[pic] |[pic] |[pic] |2 |[pic] | |90° or [pic] radians |1 |0 |Undefined |0 |Undefined |1 | |120° or [pic] radians |[pic] |[pic] |[pic] |[pic] |-2 |[pic] | |135° or [pic] radians |[pic] |[pic] |-1 |-1 |[pic] |[pic] | |150° or [pic] radians |[pic] |[pic] |[pic] |[pic] |[pic] |2 | |180° or [pic] radians |0 |-1 |0 |Undefined |-1 |Undefined | |210° or [pic] radians |[pic] |[pic] |[pic] |[pic] |[pic] |-2 | |225°or [pic] radians |[pic] |[pic] |1 |1 |[pic] |[pic] | |240° or [pic] radians |[pic] |[pic] |[pic] |[pic] |-2 |[pic] | |270° or [pic] radians |-1 |0 |Undefined |0 |Undefined |-1 | |300° or [pic] radians |[pic] |[pic] |[pic] |[pic] |2 |[pic] | |315° or [pic] radians |[pic] |[pic] |-1 |-1 |[pic] |[pic] | |330° or [pic] radians |[pic] |[pic] |[pic] |[pic] |[pic] |-2 | |360° or [pic] radians |0 |1 |0 |Undefined |1 |Undefined | |
2. APPLICATIONS OF RADIAN MEASURE
➢ ARC LENGTH OF A CIRCLE: The length of an arc is proportional to the measure of its central angle.
T
Q
s r
[pic] radians 1 radian
R r O r P
[pic]
o Arc Length: The length s of the arc intercepted on a circle of radius r by a central angle of measure [pic] radians is given by the product of the radius and the radian measure of the angle, or
[pic], [pic] in radians.
▪ Example: Find the radius of a circle in which a central angle of 2 radians intercepts an arc of length 3 feet.
[pic]
o Area of a Sector: The area of a sector of a circle of radius r and central angle [pic] is given by
[pic], [pic] in radians.
▪ Example: Find the area of a sector of a circle having a radius of 18.3 m and a central angle of 125°.
First you have to change the angle to radian measure. So we have [pic]. Now we can solve the problem using the area for a sector of a circle.
[pic]
3. CIRCULAR FUNCTIONS OF REAL NUMBERS
➢ THE CIRCULAR FUNCTIONS
o The radius (r) is always 1. Therefore, from our earlier equation (derived from the distance formula) we have [pic].
o Trigonometric functions for the circular functions
▪ Recall that that the radian measure of [pic] is related to the arc length s. Here, r = 1, so s = r[pic] gives us s = [pic].
▪ [pic] [pic] [pic]
(0, 1)
(-1, 0) (1, 0)
(0, -1)
▪ Since sin s = y and cos s = x, we can replace x and y in the equation [pic] and obtain the identity [pic].
o Domain and Range of the Circular Functions
▪ Since the ordered pair (x, y) represents a point on the unit circle,
[pic]
• The domain of cos s and sin s is the set of all real numbers, since cos s and sin s exist for any value of s
• The domains of tan s and sec s are restricted since x has a value of 0 at [pic] plus any multiple of [pic], positive or negative. So we write the domains for tan s and sec s as follows: [pic].
• The domains of cot s and csc s are restricted since y has a value of 0 at any multiple of [pic], positive or negative. So we write the domains for cot s and csc s as follows: [pic].
➢ The Unit Circle: The figure below shows the relationship between degree and radian measure. This is something you need to memorize! It will make the rest of the course much easier.
[pic]
4. LINEAR AND ANGULAR VELOCITY
➢ Linear Velocity
o In many situations we need to know how fast a point on a circle is moving or how fast the central angle is changing. Suppose that point P moves at a constant speed along a circle of radius r and center O. The measure of how fast the position of P is changing is called linear velocity. If v represents velocity, then we have
[pic]
where s is the length of the arc traced by point P at time t. This formula is just [pic] with s as the distance, v as the rate, and t is still time.
P
s
O r B
➢ Angular Velocity
o Consider the figure above. As point P moves along the circle, [pic] rotates around the origin. Since [pic] is the terminal side of [pic], the measure of the angle changes as P moves along the circle. The measure of how fast [pic] is changing is called angular velocity. Angular velocity is denoted by [pic] is given as
[pic]
where [pic] is the measure of [pic] at time t. [pic] is expressed as radians per unit of time.
▪ Recall from section 3.2 that the arc length of a circle was measured by [pic]. Now that we know the meaning of [pic], we have an alternate formula for linear velocity.
[pic].
This formula relates linear and angular velocities.
• Note that a radian is a “pure number”, with no units associated with it. This is why you get some length per unit of time with no extra unit like degrees.
o Example: Radius of a gear. A gear is driven by a chain that
travels 1.46 m/s. Find the radius of the gear if it
makes 46 rev/minute.
Solution: v = 1.46 m/s. One complete revolution is 2[pic] radians,
and 46 seconds is [pic] minute, so [pic] radians per second. Now we have only one unknown in the formula for linear velocity [pic]. Substituting for v and [pic] we have [pic]
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