Trigonometry PDF

# Trigonometry - originated from the Latin words “trigonon and metria” which means triangle and measurements - It is a branch of Mathematics that deals with the relationship of triangle, angles, and their measurements. ## Angle - An angle is a figure formed by two half lines or rays that have c...

# Trigonometry - originated from the Latin words “trigonon and metria” which means triangle and measurements - It is a branch of Mathematics that deals with the relationship of triangle, angles, and their measurements. ## Angle - An angle is a figure formed by two half lines or rays that have common end points. The common end point is called vertex and the rays are the legs. - In some other books, angle is defined as amount of space covered by the rotation of one of the rays. The fixed ray is called initial side while the moving ray is called terminal side. ## Naming an Angle 1. Using Greek Alphabets - Theta θ - Phi φ - Beta β - Omega ω - Alpha α - Gamma γ 2. English Capital Letter w/ degree symbol: - A°, B°, C°, ... 3. English Letter w/ angle sign: - ∠A, ∠B, ∠C, ... - ∠ABC ## Rotation of an Angle - **Counter-clockwise Rotation**: is the opposite direction of the movement of the hand of clock. It is represented by positive (+) direction. - **Clockwise Rotation**: is similar to the direction of the movement of the hand of a clock. It is labeled as negative (-) direction. ## Angle Measurement Units ### Degree (°) - An angle measurement taken from dividing a whole circle into 360 parts. - A unit degree can be divided further into minute and second units. - Thus, 1 degree (1°) = 60 minutes (60') - 1 minute (1') = 60 seconds (60") - 1 degree (1°) = 3600 seconds (3600") - **Degree Measurements**: Shows a circle divided into 360 degrees with markings for 0-360 degrees in 30 degree increments. ### Radian (rad) - It is the standard unit of an angle. - It is the equivalent length of the arc or the circumference of the unit circle. ## Angle Standard Position - An angle is in standard position in Cartesian plane when its initial side lies on the positive x-axis and the vertex is located at the origin. ### Example #1: Plot the following angles and identify in Cartesian Plane: 1. A° = 60° 2. β = 250° - Diagram shows an angle of 60° in the first quadrant and an angle of 250° in the third quadrant. ### Example #2: Plot the following angles and identify in Cartesian Plane: 1. ∠C = 400° 2. α = π⁄2 rad - Diagram shows an angle of 400° in the first quadrant and an angle of π⁄2 in the positive y-axis. ### Example #3: Plot the following angles and identify in Cartesian Plane: 1. 0 = -30° 2. ∠D = - π rad - Diagram shows an angle of -30° in the fourth quadrant and an angle of - π in the negative x-axis. ## Conversion Angle Units - The degree and radian units are all taken from a full circle. - Complete Circle: 360° = 2π rad - Half Circle: 180° = π rad - To convert **degrees to radian**, multiply degrees by π rad / 180 degrees. - To convert **radian to degrees**, multiply radians by 180 degrees / π rad. ### Example # 4: Convert the following angle units: 1) 240° in radian 2) 325° in radian - Answer shows work to convert 240 degrees to 4π⁄3 radians and 325 degrees to 65π⁄36 radians. ### Example # 5: Convert the following angle units: 1) 5π⁄6 rad in degree 2) 18π⁄17 rad in degree - Answer shows work to convert 5π⁄6 radians to 150° and 18π⁄17 radians to 190°35'24". ### Example # 6: Convert the following angle units: 1) 8.47 rad in degree 2) 5.78 rad in degree - Answer shows work to convert 8.47 radians to 485°32'24" and 5.78 radians to 331°20'24". ### Example # 7: Convert the following angle units: 1) 47.72° in DMS 2) 55.33° in DMS - Answer shows work to convert 47.72° to 47°43'12" and 55.33° to 55°19'33". ### Example # 8: Convert the following angle units in degrees (decimal): 1) 9°13'20" 2) 28°18'36" - Answer shows work to convert 9°13'20" to 13.24° and 28°18'36" to 28.31°. ## Relationship Angles ### Complementary Angles - Are two angles whose sum is exactly 90° or π⁄2 rad. - **Examples**: - 50° and 40° - 30° and 60° - π⁄2 rad and π⁄2 rad - 2⁄7 π rad and 10⁄14 π rad - 3⁄7 π rad and 11⁄14 π rad ### Example # 1: Get the complementary angles of the ff: 1. 60°: 90°-60° = 30° 2. 53.2°: 90°-53.2° = 36.8° 3. 2π⁄9 rad: π⁄2 - 2π⁄9 = 5π⁄18 rad 4. π⁄8 rad: π⁄2 - π⁄8 = 3π⁄8 rad ### Supplementary Angles - Are two angles whose sum is exactly 180° or π rad. - **Examples**: - 120° and 60° - 100° and 80° - π⁄2 rad and 3⁄2 π rad - 1⁄3 π rad and 2⁄3 π rad - 7⁄9 π rad and 2⁄9 π rad ### Example # 2: Get the supplementary angles of the ff: 1. 105°: 180°-105° = 75° 2. 121.9°: 180°-121.9° = 58.1° 3. 4π/13 rad: π - 4π/13 = 9π/13 rad 4. 6π/11 rad: π - 6π/11 = 5π/11 rad ## Coterminal Angles - Two angles are coterminal when they have the same initial and terminal side. The terminal side may rotate either the clockwise or counter clockwise direction. - **For counter-clockwise Direction**: - Coterminal Angles = θ + (360°) (n) - Coterminal Angles = θ + (2π) (n) - **For clockwise Direction**: - Coterminal Angles = θ - (360°) (n) - Coterminal Angles = θ - (2π) (n) - Where n is any integer, representing the number of revolution and θ is the given angle. - Diagram shows the standard position of angles in the four Quadrants. - **Quadrant I**: 0° < θ < 90° or 0 < θ < π⁄2 - **Quadrant II**: 90° < θ < 180° or π⁄2 < θ < π - **Quadrant III**: 180° < θ < 270° or π < θ < 3π⁄2 - **Quadrant IV**: 270° < θ < 360° or 3π⁄2 < θ <2π ### Example # 3: Find the coterminal sides of the following angle, (clockwise and counter clockwise) 1. 100° 2. -330° - **Counter clockwize**: * -330°+360° = 30° * -330°+360°(3) = 750° - **Clockwise**: * -330°-360° = -690° * -330°-360°(3) = -1410° ### Example # 4: Find the coterminal sides of the following angle, (clockwise and counter clockwise) 1. -π/4 rad 2. 2π/3 rad - **Counter clockwize**: * -π/4 + 2π = 7π⁄4 - **Clockwise**: * -π/4 - 2π = -9π⁄4 * 7π⁄4 - 2π = -π⁄4 * 7π⁄4 - 4π = -π⁄4 - **Counter clockwize**: * 2π/3 - 2π = -4π⁄3 * 2π/3 + 2π = 8π⁄3 - **Clockwise**: * 2π/3 - 2π = -4π⁄3 * 2π/3 - 2π(2) = -10π⁄3 ## Reference Angles - Any smallest positive acute angle that is measured from terminal side of a given angle to the nearest x-axis. - Diagrams shows the standard position of angles in the four Quadrants and the formula for the Reference Angle. - **Quadrant I**: RA = θ - **Quadrant II**: RA = 180°-θ - **Quadrant III**: RA = θ - 180° - **Quadrant IV**: RA = 360°-θ - **Radian**: - Quadrant I: RA = θ - Quadrant II: RA = π - θ - Quadrant III: RA = θ - π - Quadrant IV: RA = 2π - θ ### Example # 5: Find the reference angle of the ff: 1. 115° 2. 60° - Diagram shows the Reference Angle for 115° is 65° and the Reference Angle for 60° is 60° ### Example # 6: Find the reference angle of the ff: 1. -135° 2. 570° - Diagram shows the Reference Angle for -135° is 45° and the Reference Angle for 570° is 30°. ### Example # 7: Find the reference angle of the ff: 1. -π/6 rad 2. π rad - Diagram shows the Reference Angle for -π/6 rad is π/6 rad and the Reference Angle for π rad is 0 rad.