Trigonometry PDF

UNIT 2: TRIGONOMETRY CALCULUS 1 TRIGONOMETRY Derived from two Greek words trigonon (triangle) and metria (measurement) Branch of mathematics which deals with measurement of triangles (i.e., their sides and angles), or more specifically, with the indirect measurement of line segments a...

UNIT 2: TRIGONOMETRY CALCULUS 1 TRIGONOMETRY Derived from two Greek words trigonon (triangle) and metria (measurement) Branch of mathematics which deals with measurement of triangles (i.e., their sides and angles), or more specifically, with the indirect measurement of line segments and angles. ANGLE formed by one ray rotating about a fixed point on another ray that is stationary. The fixed point is the vertex of the angle The stationary ray is the initial side of the angle The revolving ray is the terminal side. ANGLE Use symbol ∠ to denote an angle, while the Greek letters such as 𝜃, 𝛼, 𝛽, 𝑎𝑛𝑑 𝛾 shall represent the name of the angle Angles formed by a counterclockwise rotation are considered positive angles and angles formed by a clockwise rotation are considered negative angles. An angle is said to be in standard position if its initial side is along the positive x-axis and its vertex is at the origin. The measure of an angle is determined by the amount of rotation of the initial side.  UNITS OF ANGLE MEASUREMENTS DEGREE One of the oldest and most widely used methods is to divide one revolution into 360 parts; each part is then called one degree (1°). Each degree is divided into 60 equal parts called minutes, and each minute is divided into 60 equal parts called seconds. 1 revolution = 360° 1° = 60’ 1’= 60’’  RADIAN Radian is a measure of the central angle subtended by an arc s whose length is equal to the radius r of the circle. It is denoted by rad. Angle in full rotation = 2𝜋 = 360° 𝜋 = 180°  RADIAN TO DEGREE CONVERSION (AND VICE-VERSA) !"#° To convert radians to degrees, multiply by % % To convert degrees to radians, multiply by !"#°  EXAMPLES: Convert −120° to exact radian measure Solution: 𝜋 2𝜋 −120° =− 180° 3 &% Convert to exact degree measure " Solution: 5𝜋 180° = 112.5° 𝑜𝑟 112°30′ 8 𝜋  KINDS OF ANGLES NAME ANGLE MEASURE Acute angle Between 0° 𝑎𝑛𝑑 90° Right angle Exactly 90° Obtuse angle Between 90° 𝑎𝑛𝑑 180° Straight angle Exactly 180° Reflex angle Greater than 180° but less than 360° Complementary angles 𝛼, 𝛽 𝛼 + 𝛽 = 90° Supplementary angles 𝛼, 𝛽 𝛼 + 𝛽 = 180° ACUTE ANGLE RIGHT ANGLE OBTUSE ANGLE REFLEX ANGLE  TRIANGLES A triangle is a polygon with three sides and three interior angles. The sum of the interior angles of a triangle is 180°  CLASSIFICATION OF TRIANGLES A. According to the length of its sides 1. Scalene – no sides are equal 2. Isosceles – two sides are equal 3. Equilateral – three sides are equal B. According to the measure of its angles 1. Right triangle – with one right angle 2. Oblique triangle – with no right angle a. Acute triangle – with all angles acute b. Obtuse triangle – with one obtuse angle TRIGONOMETRY OF RIGHT TRIANGLES  PYTHAGOREAN THEOREM The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. It is sed to find the side of a right triangle. Referring to the right triangle below, then  RELATIONSHIPS AMONG TRIGONOMETRIC RATIOS A. Cofunction Relationship  EXAMPLES: Find an angle 𝜃 that makes each of the statements true. (a)𝑠𝑖𝑛 𝜃 = 𝑐𝑜𝑠( 3𝜃 − 14°) (b) 𝑐𝑜𝑠 2𝜃 = 𝑠𝑖𝑛(45° + 3𝜃) SOLUTION (a) SOLUTION (b) sin 𝜃 = cos(3𝜃 − 14°) cos 2𝜃 = sin(45° + 3𝜃) sin 𝜃 90° − 2𝜃 = 45° + 3𝜃 = 90° − 3𝜃 − 14° 90° − 45° = 3𝜃 + 2𝜃 𝜃 = 90° − 3𝜃 − 14° 5𝜃 = 45° 4𝜃 = 104° 𝜃 = 9° 𝜃 = 26°  RELATIONSHIPS AMONG TRIGONOMETRIC RATIOS B. RECIPROCAL RELATIONSHIPS 𝑠𝑖𝑛 𝜃 𝑐𝑠𝑐 𝜃 = 1 𝑐𝑜𝑠 𝜃 𝑠𝑒𝑐 𝜃 = 1 𝑡𝑎𝑛 𝜃 𝑐𝑜𝑡 𝜃 = 1 C. PYTHAGOREAN RELATIONSHIPS 𝑠𝑖𝑛' 𝜃 + 𝑐𝑜𝑠 ' 𝜃 = 1 1 + 𝑡𝑎𝑛' 𝜃 = 𝑠𝑒𝑐 ' 𝜃 1 + 𝑐𝑜𝑡 ' 𝜃 = 𝑐𝑠𝑐 ' 𝜃  RELATIONSHIPS AMONG TRIGONOMETRIC RATIOS D. QUOTIENT RELATIONSHIPS sin 𝜃 tan 𝜃 = cos 𝜃 cos 𝜃 cot 𝜃 = sin 𝜃 TRIGONOMETRIC FUNCTIONS OF SPECIAL ANGLES 𝜽 𝒔𝒊𝒏 𝜽 𝒄𝒔𝒄 𝜽 𝒄𝒐𝒔 𝜽 𝒔𝒆𝒄 𝜽 𝒕𝒂𝒏 𝜽 𝒄𝒐𝒕 𝜽 30° 1 2 3 2 3 3 3 2 2 3 3 45° 2 2 2 2 1 1 2 2 60° 3 2 3 1 2 3 3 2 3 2 3  APPLICATIONS OF RIGHT TRIANGLE 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 the angle of depression of the object.  EXAMPLE From the top of a 150 ft lighthouse, the angle of depression of a boat at sea is 57º. Find the horizontal distance from the boat to the base of the lighthouse SOLUTION: 150 tan 57° = 𝑥 150 𝑥= tan 57° 𝒙 = 𝟗𝟕. 𝟒𝟏 𝒇𝒕  EXAMPLE Ryan bought a new shop and wants to order a new sign for the roof of the building. From point P, he finds the angle of elevation of the roof, from ground level, to be 31º and the angle of elevation of the top of the sign to be 42º. If point P is 24 feet from the building, how tall is the sign? SOLUTION: ℎ tan 42° = 24 ℎ = 24 tan 42° (1) ℎ−𝑦 tan 31° = 24 24 tan 31° = ℎ − 𝑦 (2) Then, substituting h: 24 tan 31° = 24 tan 42° − 𝑦 𝑦 = 24 tan 42° − 24 tan 31° 𝒚 = 𝟕. 𝟐 𝒇𝒕  BEARING AND COURSE In navigation and surveying problems, there are two commonly used methods for specifying direction. The angular direction in which the craft is pointed is called the heading. Heading is expressed in terms of an angle measured clockwise from north. The angular direction used to locate one object in relation to another object is called the bearing. Bearing is expressed in terms of the acute angle formed by a north-south line and the line of direction. As can be seen from the figure, the path from Puerto Princesa to Naga and the path from Naga to Zamboanga form a right angle at Naga. If we represent d as distance between Puerto Princesa and Zamboanga then by Pythagorean Theorem: 𝑑 = 65! + 130! = 65 5 𝑜𝑟 𝑎𝑝𝑝𝑟𝑜𝑥 145.3𝑚𝑖𝑙𝑒𝑠  COORDINATE PLANE The coordinate axes divide the plane into four parts called quadrants. For any given angle in standard position, the measurement boundaries for each quadrant are summarized as follows:  TRIGONOMETRIC FUNCTIONS OF ANY ANGLE Let P(x, y) be any point, except the origin, on the terminal side of an angle θ in standard position. Let r = d(O, P), the distance from the origin to P. The six trigonometric functions of θ are:  TRIGONOMETRIC FUNCTIONS OF QUADRANTAL ANGLES A quadrantal angle is an angle whose terminal side coincides with the x- or y-axis. The value of a trigonometric function of a quadrantal angle can be found by choosing any point on the terminal side of the angle and then applying the definition of that trigonometric function.  When an angle lies along an axis, the values of the trigonometric functions are either 0, 1, -1, or undefined. When the value of a trigonometric function is undefined, it means that the ratio for that given function involved division by zero. Below is a table with the values of the functions for quadrantal angles. 𝜽 𝟎 𝟗𝟎° 𝟏𝟖𝟎° 𝟐𝟕𝟎° sin 𝜃 0 1 0 −1 cos 𝜃 1 0 −1 0 tan 𝜃 0 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 0 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 csc 𝜃 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 0 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 0 sec 𝜃 1 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 −1 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 cot 𝜃 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 1 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 −1  REFERENCE ANGLE The reference angle is the acute angle that the given angle in standard position makes with the x-axis. In other words, the reference angle measures the closest distance of that terminal side of the given angle to the x-axis.  EXAMPLES Determine the reference angle for the given angle. a) 𝟐𝟑𝟏° b) 𝟏𝟒𝟎𝟔° Solutions: a) This angle terminates at the third quadrant, hence the reference angle = 231° − 180° = 51° b) This angle terminates at the fourth quadrant coinciding with the terminal sides of 326°. Hence, the reference angle = 360 − 326° = 34°  EXAMPLE Find the exact value of the six trigonometric functions of 495° Solution: The given angle terminates at the same terminal side of 135°, hence its reference angle = 180° − 135°. Taking now the trigonometric functions of 45° at the second quadrant we have, 2 sin 495° = sin 45° = 2 2 cos 495° = − cos 45° = − 2 tan 495° = − tan 45° = −1 2 csc 495° = csc 45° = = 2 1 sec 495° = − sec 45° = − 2 cot 495° = − cot 45° = −1  CIRCULAR FUNCTIONS A circle with center at the origin and radius 1 is called a unit circle. The equation of the unit circle is 𝑥 ] + 𝑦 ] = 1  TRIGONOMETRIC IDENTITIES An equation involving a variable is called an identity if equality holds for every value of the variable for which all terms in the equation are defined. Tips in proving identities: 1. Start with the more complicated side of the identity and transform it into the simpler one. 2. Try using basic or other known identities. 3. Try algebraic operations such as multiplying, factoring, combining fractions, or splitting fractions. 4. If other steps fail, try expressing each function in terms of sine and cosine functions; then perform appropriate algebraic operations. 5. At each step, keep the other side of the identity in mind. This often reveals what you should do in order to get there.  EXAMPLES ; !?@A; > Simplify the expression + !?@A; > ; Prove each of the identities: ;? @A; > ; = ;B@ >? @;@ > ;B@ > CDE ! >?FGC ! > = 1 − sin 𝜃 cos 𝜃 ;?@A; > NEGATIVE ANGLE IDENTITIES sin −𝜃 = − sin 𝜃 cos −𝜃 = cos 𝜃 tan(−𝜃) = − tan 𝜃 csc −𝜃 = − csc 𝜃 sec −𝜃 = sec 𝜃 cot −𝜃 = − cot 𝜃  ADDITION FORMULAS sin 𝐴 ± 𝐵 = sin 𝐴 cos 𝐵 ± cos 𝐴 sin 𝐵 cos 𝐴 ± 𝐵 = cos 𝐴 cos 𝐵 ∓ sin 𝐴 sin 𝐵 tan 𝐴 ± tan 𝐵 tan(𝐴 ± 𝐵) = 1 ∓ tan 𝐴 tan 𝐵  EXAMPLES Determine the exact value by expressing the given angle in terms of the sum or difference of two special angles. cos 15° cot 105°  EXAMPLES Evaluate without using calculator sin 12° cos 33° + cos 12° sin 33° HI= !&°JHI= K&° !?HI= !&° HI= K&°  EXAMPLES Find sin(𝐴 + 𝐵) and cos(𝐴 − 𝐵) if: L & sin 𝐴 = , A in Q2 and cos 𝐵 = − , B in Q3 & !L DOUBLE ANGLE AND HALF-ANGLE FORMULAS  EXAMPLES K Ø If sin 𝜃 = − '&, 𝜃 is in Q3. Find the following: sin 2𝜃 > cos ' ;