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MODULE 3 - TRIGONOMETRIC FUNCTIONS
OVERVIEW
Trigonometry (Greek word trigonon triangle and metron measure) is the branch of mathematics that deals with the measurement of sides and angles of a triangle. Trigonometry has many applications in engineering, surveying, construction, and navigation. This module will be discussing the topics about radians and degrees, trigonometric functions, inverse trigonometric functions, and the graph of sine and cosine functions.
Trigonometry (Greek word trigonon triangle and metron measure) is the branch of mathematics that deals with the measurement of sides and angles of a triangle. Trigonometry has many applications in engineering, surveying, construction, and navigation. This module will be discussing the topics about radians and degrees, trigonometric functions, inverse trigonometric functions, and the graph of sine and cosine functions.
LEARNING OBJECTIVES:
At the end of this module, you should be able to:1. Convert between degrees and radians.2. Solve trigonometric functions of any angle and solve real-life problems.3. Determine the amplitude, period, and cycle of a sine and cosine graphs.4. Sketch the graphs of sine and cosine.
At the end of this module, you should be able to:1. Convert between degrees and radians.2. Solve trigonometric functions of any angle and solve real-life problems.3. Determine the amplitude, period, and cycle of a sine and cosine graphs.4. Sketch the graphs of sine and cosine.
PLANE ANGLES
A plane angle is formed by two rays meeting at a common endpoint. Shown in the figure, rays OB and OA formed the plane angle BOA. Rays OB and OA are the sides of the angle and the point O is the vertex. Further, a plane angle can also be made by revolving a ray, either clockwise or counterclockwise, from the initial side to the terminal side. If the direction of rotation is counterclockwise, the angle is positive. If the direction of rotation is clockwise, the angle is negative.
A plane angle is formed by two rays meeting at a common endpoint. Shown in the figure, rays OB and OA formed the plane angle BOA. Rays OB and OA are the sides of the angle and the point O is the vertex. Further, a plane angle can also be made by revolving a ray, either clockwise or counterclockwise, from the initial side to the terminal side. If the direction of rotation is counterclockwise, the angle is positive. If the direction of rotation is clockwise, the angle is negative.
MEASURES OF AN ANGLE
A degree (°) is defined as the measure of the central angle subtended by an arc of a circle equal to 1/360 of the circle's circumference. A minute ( ′ ) is 1/60 of a degree. A second ( ′′ ) is 1/60 of a minute, or 1/3600 of a degree. More so, a radian (rad) is defined as the measure of the central angle subtended by an arc of a circle equal to the circle's radius (Moyer). A circle has a circumference of 2π(radius) and the angle subtended is 360°. Thus, 1 radian = 180°/π or 57.296°. To sum up:1 minute ( ′ ) = 1/60°1 second ( ′′ ) = 1/60 ′ or 1/3600°1 radian = 180°/π or 57.296°
Moreover, in converting degrees to radians, multiply the degree measure by π/180° and in converting radians to degrees, multiply the radian measure by 180°/π.
A degree (°) is defined as the measure of the central angle subtended by an arc of a circle equal to 1/360 of the circle's circumference. A minute ( ′ ) is 1/60 of a degree. A second ( ′′ ) is 1/60 of a minute, or 1/3600 of a degree. More so, a radian (rad) is defined as the measure of the central angle subtended by an arc of a circle equal to the circle's radius (Moyer). A circle has a circumference of 2π(radius) and the angle subtended is 360°. Thus, 1 radian = 180°/π or 57.296°. To sum up:1 minute ( ′ ) = 1/60°1 second ( ′′ ) = 1/60 ′ or 1/3600°1 radian = 180°/π or 57.296°
Moreover, in converting degrees to radians, multiply the degree measure by π/180° and in converting radians to degrees, multiply the radian measure by 180°/π.
STANDARD POSITION
An angle is in standard position when the vertex is at the origin, and the initial side is along the positive x-axis. The ray on the x-axis is called the initial side and the other ray is called the terminal side. If the terminal side of an angle lies "on" the axes (such as 0º, 90º, 180º, 270º, 360º ), it is called a quadrantal angle. The angle shown at the left is referred to as a Quadrant II angle since its terminal side lies in Quadrant II.
An angle is in standard position when the vertex is at the origin, and the initial side is along the positive x-axis. The ray on the x-axis is called the initial side and the other ray is called the terminal side. If the terminal side of an angle lies "on" the axes (such as 0º, 90º, 180º, 270º, 360º ), it is called a quadrantal angle. The angle shown at the left is referred to as a Quadrant II angle since its terminal side lies in Quadrant II.
COTERMINAL ANGLES
Two angles which, when placed in standard position, have coincident terminal sides are called coterminal angles (Moyer). In the diagram at the right, +120º and -240º are coterminal angles. For example, to solve two positive angles that are coterminal with 50°, we will add 360° and then add 360° again.Positive Coterminal Angle 1 = 50° + 360° = 410°Positive Coterminal Angle 2 = 50° + 360° + 360° = 770°
Two angles which, when placed in standard position, have coincident terminal sides are called coterminal angles (Moyer). In the diagram at the right, +120º and -240º are coterminal angles. For example, to solve two positive angles that are coterminal with 50°, we will add 360° and then add 360° again.Positive Coterminal Angle 1 = 50° + 360° = 410°Positive Coterminal Angle 2 = 50° + 360° + 360° = 770°
TRIGONOMETRIC FUNCTIONS OF AN ANGLE
A point P, plotted in the Cartesian plane, with coordinate (x,y) is shown in the figure. Angle θ is in standard position considering a line connecting the point P to the origin and the positive x-axis. The distance of the point from the origin is r and r= √(x2 + y2) ≠ 0. This figure shows a triangle with a 90° angle (right triangle).
A point P, plotted in the Cartesian plane, with coordinate (x,y) is shown in the figure. Angle θ is in standard position considering a line connecting the point P to the origin and the positive x-axis. The distance of the point from the origin is r and r= √(x2 + y2) ≠ 0. This figure shows a triangle with a 90° angle (right triangle).
TRIGONOMETRIC FUNCTION VALUES
To be able to find the trigonometric function values of a certain problem, there is two mnemonic devices to be familiarized namely, the SOHCAHTOA and the CHOSHACAO. They can help students in remembering the six trigonometric functions of theta as well as their corresponding formula on computation.
For instance, let us solve this problem from problem set #3: One end of a foot ruler is placed against a vertical wall; the other end of the reaches a point on the floor 9 inches from the base of the wall. Find the sine, cosine, and tangent of the angle that the ruler makes: a. With the wall; b. With the floor.
To be able to find the trigonometric function values of a certain problem, there is two mnemonic devices to be familiarized namely, the SOHCAHTOA and the CHOSHACAO. They can help students in remembering the six trigonometric functions of theta as well as their corresponding formula on computation.
For instance, let us solve this problem from problem set #3: One end of a foot ruler is placed against a vertical wall; the other end of the reaches a point on the floor 9 inches from the base of the wall. Find the sine, cosine, and tangent of the angle that the ruler makes: a. With the wall; b. With the floor.
The first thing I did is visualize the problem description by creating a simple drawing as shown at the left. There are two given quantities: the one-foot ruler and the distance from the base of the wall which is 9 inches. To get the length of the wall occupied by the ruler, I just used the Pythagorean Theorem which states that "the sum of the square of the legs of a right triangle is equal to the square of the hypotenuse" or c2 = a2 + b2 where c is the hypotenuse and a and b are the legs. Once the three sides of the triangle are revealed, we can now start computing the six trigonometric functions while taking in consideration where theta is located. In letter a, theta is the angle that the ruler makes with the wall while in letter b, theta is the angle that the ruler makes with the floor. Finally, we will now use the SOHCAHTOA to find the sine, cosine, and tangent of the angle. Note that the answer should be in lowest term and is simplified.
INVERSE TRIGONOMETRIC FUNCTIONS
The inverse functions of the basic trigonometric functions, sine, cosine, tangent, cotangent, secant, and cosecant functions, are known as inverse trigonometric functions. Arcus functions, anti-trigonometric functions, and cyclometric functions are all terms used to describe them. In trigonometry, these inverse functions are used to get the angle with any of the trigonometric ratios. In engineering, physics, geometry, and navigation, inverse trigonometry functions are extremely useful.
The inverse functions of the basic trigonometric functions, sine, cosine, tangent, cotangent, secant, and cosecant functions, are known as inverse trigonometric functions. Arcus functions, anti-trigonometric functions, and cyclometric functions are all terms used to describe them. In trigonometry, these inverse functions are used to get the angle with any of the trigonometric ratios. In engineering, physics, geometry, and navigation, inverse trigonometry functions are extremely useful.
Let us take this example from my activity 3. To get the approximate value of the angle in degrees, we simply find the inverse of the trigonometric function. We may enter the data into the scientific calculator while considering whether the angle should be in degrees or radians depending on what is asked.
ANGLE OF ELEVATION AND DEPRESSION
The angle of elevation of an object which is above the eye of an observer is the angle which the line of sight to the object makes with the horizontal. If the object is below the eye of the observer, the angle which the line of sight makes with the horizontal is called the angle of depression of the object (Rider).
Note: "An angle of elevation of one location relative to another is always congruent (equal in measure) to the angle of depression of the first location relative to the second."
The angle of elevation of an object which is above the eye of an observer is the angle which the line of sight to the object makes with the horizontal. If the object is below the eye of the observer, the angle which the line of sight makes with the horizontal is called the angle of depression of the object (Rider).
Note: "An angle of elevation of one location relative to another is always congruent (equal in measure) to the angle of depression of the first location relative to the second."
ANGLE OF ELEVATION
This example is from problem set #3: At a point 160 feet from a building, and in a horizontal line with its base, the angle of elevation of the top of the building is 37°. How high is the building? By making an illustration, we can infer two given quantities namely, the distance between the point and the building (160 feet) and the angle of elevation (37°). To find the height of the building, we will use tangent since relative to the angle, we are dealing with adjacent and opposite. By simplifying the equation, we can get x or the height of the building which is 120.6 feet.
This example is from problem set #3: At a point 160 feet from a building, and in a horizontal line with its base, the angle of elevation of the top of the building is 37°. How high is the building? By making an illustration, we can infer two given quantities namely, the distance between the point and the building (160 feet) and the angle of elevation (37°). To find the height of the building, we will use tangent since relative to the angle, we are dealing with adjacent and opposite. By simplifying the equation, we can get x or the height of the building which is 120.6 feet.
ANGLE OF DEPRESSION
Example from quiz 3: The angle of depression from the top of a cliff to point A on the ground is 35 degrees. If point A is 280 feet from the base of the cliff, how tall is the cliff? As stated above, the angle of depression and the angle of elevation must be congruent because the hypotenuse of the supposed triangle will act as transversal of two parallel lines (horizontal), thus the alternate interior angles must be congruent. Since we are now dealing with opposite and adjacent relative to the angle of elevation, I used tangent to solve the height of the cliff. Henceforth, the cliff is 196 feet high.
Example from quiz 3: The angle of depression from the top of a cliff to point A on the ground is 35 degrees. If point A is 280 feet from the base of the cliff, how tall is the cliff? As stated above, the angle of depression and the angle of elevation must be congruent because the hypotenuse of the supposed triangle will act as transversal of two parallel lines (horizontal), thus the alternate interior angles must be congruent. Since we are now dealing with opposite and adjacent relative to the angle of elevation, I used tangent to solve the height of the cliff. Henceforth, the cliff is 196 feet high.
GRAPHS OF SINE AND COSINE
The sine and cosine functions have several distinct characteristics:
The sine and cosine functions have several distinct characteristics:
- They are periodic functions with a period of 2π.
- The domain of each function is (−∞,∞) and the range is [−1,1].
- The graph of y = sin x is symmetric about the origin, because it is an odd function.
- The graph of y = cos x is symmetric about the y-axis, because it is an even function.
THE SINE GRAPH
To draw the equation y = sinx, assume values of x, and substitute the value of x in the equation to determine the value of y. (Tip: Your calculator must be in RADIAN mode since the given values of x are in radians.) For this curve, we observe that the highest point in the graph is 1 and the lowest is -1. The curve extends to both +∞ and −∞. The length of one complete cycle (period) of the curve is 2π. It crosses the y-axis at the origin (0,0) and at (nπ,0) for the x-intercepts.
To draw the equation y = sinx, assume values of x, and substitute the value of x in the equation to determine the value of y. (Tip: Your calculator must be in RADIAN mode since the given values of x are in radians.) For this curve, we observe that the highest point in the graph is 1 and the lowest is -1. The curve extends to both +∞ and −∞. The length of one complete cycle (period) of the curve is 2π. It crosses the y-axis at the origin (0,0) and at (nπ,0) for the x-intercepts.
THE COSINE GRAPH
To draw the equation y = cosx, assume values of x and substitute the value of x in the equation to determine the value of y. For this curve, we observe that the highest point in the graph is 1 and the lowest is -1. The curve extends to both +∞ and −∞. The length of one complete cycle (period) of the curve is 2π. It crosses the y-axis at the origin(0,1) and at (π/2+nπ,0) for the x-intercepts.
To draw the equation y = cosx, assume values of x and substitute the value of x in the equation to determine the value of y. For this curve, we observe that the highest point in the graph is 1 and the lowest is -1. The curve extends to both +∞ and −∞. The length of one complete cycle (period) of the curve is 2π. It crosses the y-axis at the origin(0,1) and at (π/2+nπ,0) for the x-intercepts.
HOW TO GRAPH SINE FUNCTION?
The amplitude of y = A sin bx represents half the distance between the maximum and minimum values of the function and is given by: Amplitude = |A|. The period of the sine curve is the length of one cycle of the curve and is given by: Period = 2π/b.
Before anything else, one must know the amplitude as well as the period of the function to be able to graph accurately. The amplitude is 3, meaning the highest and lowest values are 3 and -3 respectively. The period is π, meaning one complete cycle is π. As previously stated, we will assume values of x and substitute the value of x in the equation to determine the y. After getting values, we can now plot it in the graph while taking in consideration the amplitude. As you can see in the graph, the value of π/6 and π/3 is 2.6 but we know that the amplitude is 3 so I extended the graph to its halfway to reach the 3. After completing one period, we can now stop graphing the function unless otherwise stated to graph at least 2 periods for instance.
The amplitude of y = A sin bx represents half the distance between the maximum and minimum values of the function and is given by: Amplitude = |A|. The period of the sine curve is the length of one cycle of the curve and is given by: Period = 2π/b.
Before anything else, one must know the amplitude as well as the period of the function to be able to graph accurately. The amplitude is 3, meaning the highest and lowest values are 3 and -3 respectively. The period is π, meaning one complete cycle is π. As previously stated, we will assume values of x and substitute the value of x in the equation to determine the y. After getting values, we can now plot it in the graph while taking in consideration the amplitude. As you can see in the graph, the value of π/6 and π/3 is 2.6 but we know that the amplitude is 3 so I extended the graph to its halfway to reach the 3. After completing one period, we can now stop graphing the function unless otherwise stated to graph at least 2 periods for instance.
HOW TO GRAPH COSINE FUNCTION?
The amplitude of y = A cos bx represents half the distance between the maximum and minimum values of the function and is given by: Amplitude = |A|. The period of the cosine curve is the length of one cycle of the curve and is given by: Period = 2π/b.
Just like above, one must know the amplitude as well as the period of the function to be able to graph accurately. The amplitude is 1/2, meaning the highest and lowest values are 0.5 and -0.5 respectively. The period is 2π/3, meaning one complete cycle is 2π/3. As previously stated, we will assume values of x and substitute the value of x in the equation to determine the y. After getting values, we can now plot it in the graph while taking in consideration the amplitude. After completing one period, we can now stop graphing the function unless otherwise stated to graph at least 2 periods for instance. But take note that both the curves of sine and cosine functions extends to positive and negative infinity.
The amplitude of y = A cos bx represents half the distance between the maximum and minimum values of the function and is given by: Amplitude = |A|. The period of the cosine curve is the length of one cycle of the curve and is given by: Period = 2π/b.
Just like above, one must know the amplitude as well as the period of the function to be able to graph accurately. The amplitude is 1/2, meaning the highest and lowest values are 0.5 and -0.5 respectively. The period is 2π/3, meaning one complete cycle is 2π/3. As previously stated, we will assume values of x and substitute the value of x in the equation to determine the y. After getting values, we can now plot it in the graph while taking in consideration the amplitude. After completing one period, we can now stop graphing the function unless otherwise stated to graph at least 2 periods for instance. But take note that both the curves of sine and cosine functions extends to positive and negative infinity.
Jess Marck Salazar -PS3- BSCE1B.pdfPROBLEM SET #3 (43/45)
JessMarckSalazar-Activity3-BSCE1B.pdfACTIVITY #3 (100/100)
Salazar-Quiz3-BSCE1B.pdfQUIZ #3 SOLUTIONS (20/20)
REFLECTION
In this module, I learned many things about trigonometric functions. First is the CHOSHACAO. It was my first time to hear this mnemonic. All I know was the SOHCAHTOA introduced by my high school teachers. Though I already know the formulas of cosecant, secant, and cotangent since then, the new mnemonic didn't come to my mind. Second one was the angles of elevation and depression. I was practiced by so many problems involving the topic and I really enjoyed them. Last would be the graph of the sine and cosine functions. It was satisfying to graph a certain function knowing that you are already familiar with how it works and how it is made. I learned how to identify a sine graph from the cosine one by observing which the graph is symmetric about. If it is symmetric with the origin, it is a sine function and if it is symmetric with the y-axis, it is a cosine function. Overall, I was able to understand the whole module. Thanks to the materials provided as well as the explanation of our teacher.
In this module, I learned many things about trigonometric functions. First is the CHOSHACAO. It was my first time to hear this mnemonic. All I know was the SOHCAHTOA introduced by my high school teachers. Though I already know the formulas of cosecant, secant, and cotangent since then, the new mnemonic didn't come to my mind. Second one was the angles of elevation and depression. I was practiced by so many problems involving the topic and I really enjoyed them. Last would be the graph of the sine and cosine functions. It was satisfying to graph a certain function knowing that you are already familiar with how it works and how it is made. I learned how to identify a sine graph from the cosine one by observing which the graph is symmetric about. If it is symmetric with the origin, it is a sine function and if it is symmetric with the y-axis, it is a cosine function. Overall, I was able to understand the whole module. Thanks to the materials provided as well as the explanation of our teacher.
REFERENCES:
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