2Q_Week 2 - Day 1_Introduction to Trigonometry.pptx

Introduction to Trigonometry
Definition and Scope: Trigonometry is introduced as the branch of mathematics that deals with the study of angles, triangles, and the relationships between their sides.
Applications: Lectures highlight the importance of trigonometry in various fields such as architecture, engineering, physics, and astronomy.
2. Basic Trigonometric Functions
Sine (sin), Cosine (cos), Tangent (tan): Defined as ratios of sides in a right-angled triangle.
Sine: Opposite side / Hypotenuse
Cosine: Adjacent side / Hypotenuse
Tangent: Opposite side / Adjacent side
Reciprocal Functions: Cosecant (csc), Secant (sec), and Cotangent (cot) are introduced.
Cosecant: Hypotenuse / Opposite side
Secant: Hypotenuse / Adjacent side
Cotangent: Adjacent side / Opposite side
3. Unit Circle and Angles
Unit Circle: Explains how trigonometric functions can be extended to all real numbers using the unit circle.
Degrees and Radians: Conversion between degrees and radians is emphasized, and angles beyond 0° to 90° are introduced.
Quadrants: Explanation of angle measurements in different quadrants and their corresponding trigonometric function signs.
4. Trigonometric Identities
Pythagorean Identity:
sin
⁡
2
𝜃
+
cos
⁡
2
𝜃
=
1
sin
2
θ+cos
2
θ=1
Sum and Difference Formulas: For sine, cosine, and tangent (e.g.,
sin
⁡
(
𝐴
+
𝐵
)
=
sin
⁡
𝐴
cos
⁡
𝐵
+
cos
⁡
𝐴
sin
⁡
𝐵
sin(A+B)=sinAcosB+cosAsinB).
Double-Angle and Half-Angle Formulas: These are used for solving more complex trigonometric problems.
Even-Odd Identities: Shows how trigonometric functions behave for positive and negative angles (e.g.,
sin
⁡
(
−
𝜃
)
=
−
sin
⁡
(
𝜃
)
sin(−θ)=−sin(θ)).
5. Solving Right Triangles
Using Trigonometric Ratios: Application of trigonometric functions to solve for unknown sides and angles in right triangles.
Inverse Trigonometric Functions: Inverse functions (arcsin, arccos, arctan) are introduced to find angles when sides are known.
6. Trigonometric Equations
Simple Equations: Basic trigonometric equations such as
sin
⁡
𝜃
=
0.5
sinθ=0.5, and how to solve them using inverse functions.
Complex Equations: Techniques to solve more complex equations involving multiple angles or identities.
7. Graphing Trigonometric Functions
Sine, Cosine, and Tangent Graphs: Understanding the shape, period, and amplitude of these functions.
Transformations: Shifts, stretches, and reflections of trigonometric function graphs.
Periodic Nature: Highlighting how these functions repeat over intervals.
8. Applications of Trigonometry
Real-World Problems: Lectures would typically cover solving practical problems like measuring heights and distances using angles of elevation and depression.
Harmonic Motion: Using sine and cosine functions to model periodic phenomena such as sound waves or tides.
9. Law of Sines and Law of Cosines
Law of Sines: Helps solve any triangle (not just right triangles), using the formula
sin
⁡
𝐴
𝑎
=
sin
⁡
𝐵
𝑏
=
sin
⁡
𝐶
𝑐
a
sinA
​
=
b
sinB
​
=
c
sinC
​
.
Law of

1. Week 2 – Day 1 Introduction to Trigonometry

2. Subject Overview Course outline Requirements 2 Reminders

3. Course Outline Unit Circle 3 Angular Measurements Circular Functions/Trigonometric Functions Graphs of Circular Functions Angles of Elevation and Depression Trigonometric Identities Trigonometric Equations

4. LESSON OBJECTIVES Illustrate the unit circle, radian measure and the intercepted arc of a central angle Discuss the relationship between the linear and angular measures of a central angle 4

5. 5 Review the Parts Write an example of each of the following: Radius ____ Secant ____ Tangent ____ Chord ____ Diameter ____ Minor arc ____ Major arc ____ Central angle ____ Inscribed angle ____ Center ____

6. 6 Relationship between the linear and angular measures of a central angle Linear Measure : The linear measure of a central angle is the length of the arc it subtends on the circumference of the circle. This length is directly proportional to the angle's measure in radians.

7. 7 Relationship between the linear and angular measures of a central angle Angular Measure : The angular measure of a central angle in a unit circle is given in radians. Radians are a way to measure angles based on the radius of the circle.

8. 8 ARC LENGTH Where is in radian, and r as radius   Arc Length is a portion of the circumference of the circle. - angle    

9. 9 ARC LENGTH Where is in radian   - angle       Minor Arc- an arc that is less than half of a circle. Major Arc - an arc that is more than half of a circle.

10. 10 TRI-GONIA-METREIN

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21. Angular Measurements Sexagesimal System Circles are divided into 360 equal parts, each being a degree. Each of these degrees can be evenly divided into 60 equal parts. These parts are called minutes. These minutes can be evenly divided into 60 equal parts. These parts are called seconds. Circular System is based on the constant relation that exists between the circumference of a circle and the radius of that circle . Radian is used as a unit of measure. A radian is an angle subtended at the center of a circle by an arc whose length is equal to the radius. 22

22. 23 Because 2  radians corresponds to one complete revolution, degrees and radians are related by the equations 360  = 2  rad and 180  =  rad.

23. Examples: Convert the following Degree to Radian and Radian to Degree 1. 27   24 Answer :   2.   Answer :   3.   Answer :   4.   Answer :   5.   Answer :