TRIGONOMETRY

TRIGONOMETRY. Performance Task No. 4. 11 th Grade.

01. 03. 02. TABLE OF CONTENTS. ANGLE MEASURE. ARC LENGTH AND AREA OF A SECTOR.

01. 02. TABLE OF CONTENTS. CIRCULAR FUNCTIONS. REFERENCE ANGLES.

ANGLE MEASURE. 01.

Angle Measure : ILLUSTRATING THE UNIT CIRCLE. The Unit Circle is basically a visual representation of certain “special angles”, for which the exact values of the trig functions are known. It is called the “unit” circle, since its radius is 1. In a unit circle, The length of the intercepted arc is equal to the radian measure of the central angle. Let (x, y) be the endpoint on the unit circle of an arc of arc lengths. The (x, y) coordinates of this point can be described as functions of the angle..

Illustrating The Relationship of the Linear and Angular Measures of a Central Angle in a Unit Circle.

Illustrating the Relationship between the Linear and Angular Measures of a Central Angle in unit Circle.

Illustrating the Relationship between the Linear and Angular Measures of a Central Angle in Unit Circle.

Convert Degree Measure to Radian Measure and Vise Versa.

Convert Degree Measure to Radian Measure and Vise Versa.

Convert Degree Measure to Radian Measure and Vise Versa.

COTERMINAL ANGLES. 02.

Coterminal Angles -Two angles in standard position that have a common terminal side are called coterminal. Observed that the degree measures of coterminal angles differ by multiples of 360°.

Key Point: Since we know that one rotation around the circle is 360 degrees, finding coterminal angles is as easy as adding or subtracting multiples of 360 to each angle. Example: Find the coterminal angles 1059°, 624°, 398° and its quadrant?.

ARC LENGTH AND AREA OF A SECTOR. 03. c.

I . Arc Length:. An Arc Length is the distance between two points along the section of a curve. In a circle of a radius r , the length of s of an arc intercepted by a central angle with measure θ radians is given by: In a circle, a central angle whose radian measure is θ subtends an arc that is the fraction theta/2pi of the circumference of the circle. Thus in a circle of a radius r (See figure of circle below) the length of s of an arc that subtends the angle θ is:.

II. Arc Length. Exmaple: Find the length of an arc of a circle with radius 10m that subtends a central angle of 30°. Solution: Since the given central angle is in degrees, we have to convert it into radiance measure. The apply the formula for an arc length. 30(π/180) = π/6 rad s = 10 (π/6) = 5π/3 m.

III. Arc Length. Example: Find the arc length Solution: S= θ/360° x 2 π r = 80°/360° x 2(3.14)(12) = 0.2222 x 75.36 = 16.74666667 = 16.8 cm.

I.Area of a Sector:. The area of a sector of a circle is the amount of space enclosed within the buondary of a sector. In a circle of a radius r, the are A of a sector with a central angle measuring is: A sector of a circle is the portion of the interior of a circle bounded by the initial and terminal sides of a central angle and its intercepted arc. Note that any angle with measure 2pi radians will define a sector of the whole circle. Therefore, if a central angle of a sector has measure θ radians, then the sector makes up the fraction theta/2pi of a complete circle..

II. Area of a Sector. (See figure of the circle) Since the area of a complete circle with radius r is pir² we have: Example: Find the area of a sector of a circle with central angle 60° if the radius of the circle is 3m. Solution: First, we have to convert 60° into radians. Then apply the formula for computing the area of a sector. 60(π/180) = π/3 rad A = ½(3²)π/3 = 3π/2 m².

III. Area of a Sector. Example: Find the area of a sector. Solution: A = θ/360° x πr² = 80°/360° x (3.14)(12)² = 0.22222 x 452.16 = 100.48 cm².

ILLUSTRATING THE DIFFERENT CIRCULAR FUNCTIONS. 04.

Illustrating the Different Circular Functions SINE: the ratio between the long side (called the hypotenuse) and the side that is opposite to an acute angle in a right triangle. COSINE: the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. TANGENT: a straight line or plane that touches a curve or curved surface at a point, but if extended does not cross it at that point. COSECANT: the ratio of the hypotenuse (in a right-angled triangle) to the side opposite an acute angle; the reciprocal of sine. SECANT: the ratio of the hypotenuse to the shorter side adjacent to an acute angle (in a right-angled triangle); the reciprocal of a cosine. COTANGENT: (in a right-angled triangle) the ratio of the side (other than the hypotenuse) adjacent to a particular acute angle to the side opposite the angle..

EXAMPLE FINDING THE VALUE OF R FORMULA: r r= √X 2 + Y 2.

DETERMINE THE 6 TRIGONOMICAL FUNCTIONS IF RADIUS IS NOT EQUAL TO 1.

REFERENCE ANGLES. 05.

Reference Angle The reference angle is the angle that the given angle makes with the x-axis. Regardless of where the angle ends, the reference angle measures the closest distance of the terminal side to the x-axis. Let’s give an example to better understand..

Example: Find the six trigonometric functions of the angle θ if the terminal side of θ in standard position passes through the point (5, −12). Let’s first identify what is the six trigonometric functions. So, we have: sin θ = y/r, also referred to as opposite side/hypotenuse cos θ = x/r, also referred to as adjacent side/hypotenuse tan θ = y/x, also referred to as opposite side/adjacent side csc θ = r/y, also referred to as hypotenuse/opposite side sec θ =r/x, also referred to as hypotenuse/adjacent side cot θ =x/y, also referred to as adjacent side/opposite side.

Now, we need to get the x and y, as well as the hypotenuse in order to find the trigonometric functions. Since the point is (5,-12), therefore the x=5 and y=-12. Let’s use the formula r²=x²+y² to find the hypotenuse. If we substitute it, it will become r=√(5)²+(-12)²=13 This time, let’s find the six trigonometric functions. Since cos θ = x/r, therefore the cos θ=5/13 Since sin θ = y/r, therefore the sin θ=-12/13 Since tan θ = y/x, therefore the tan θ=-12/5 Since sec θ = r/x, therefore the sec θ=13/5 Since csc θ = r/y, therefore the csc θ=-13/12 Since cot θ=x/y, therefore the cot θ=-5/12 And that’s how we can find the six trigonometric functions..

THANK YOU. GROUP 4. REFERENCES: https://mathhints.com/the-unit-circle/ https://m.youtube.com/watch?v=AV_RGFNc9zM https://www.cliffsnotes.com › trigonometry › circular-f... https://www.mathsisfun.com/sine-cosine-tangent.html https://courses.lumenlearning.com/csn-precalculus/chapter/reference-angles/ https://youtu.be/6BRtPfofXog https://youtu.be/cPLLf-GvF8E https://m.youtube.com/watch?v=dz5YpNFuhRQ&t=309s https://m.youtube.com/watch?v=cPLLf-GvF8E Pre-Calc LM SHS v.1.pdf PDF.