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Trigonometric Functions of Acute Angles ο Let us consider the right triangle ABC, with right angle at C. Angles A and B are acute angles which are complementary (A + B = 90°). The sides opposite angles A, B, and C will be denoted by the corresponding small letters a, b, and c, respectively. Then by taking ratios of the sides of the triangle, we define the three trigonometric functions of acute angle A as follows: sine A = cosine A = tangent A = π πππ πππππ ππ‘π π΄ βπ¦πππ‘πππ’π π π πππ ππππππππ‘ π‘π π΄ βπ¦πππ‘πππ’π π π πππ πππππ ππ‘π π΄ π πππ ππππππππ‘ π‘π π΄ ο By using the abbreviations of the three trigonometric functions and the corresponding sides of the triangle with reference to angle A, the corresponding equations are: sine A = π πππ πππππ ππ‘π π΄ βπ¦πππ‘πππ’π π cosine A = π πππ ππππππππ‘ π‘π π΄ βπ¦πππ‘πππ’π π tangent A = π πππ πππππ ππ‘π π΄ π πππ ππππππππ‘ π‘π π΄ π sin A = π π cos A = π π tan A = π ο The other three trigonometric functions that can be derived from triangle ABC are as follows with the corresponding abbreviations. cosecant A = secant A = βπ¦πππ‘πππ’π π π πππ πππππ ππ‘π π΄ βπ¦πππ‘πππ’π π π πππ ππππππππ‘ π‘π π΄ cotangent A = π πππ ππππππππ‘ π‘π π΄ π πππ πππππ ππ‘π π΄ π csc A = π π sec A = π π cot A = π In summary, acute angles (angle A & angle B) are the reference angles of the right triangle: ο We use the symbol ΞΈ (Greek letter βthetaβ) for these acute angles or capital letters A or B. Thus, ΞΈ = A or ΞΈ = B. ο π sin ΞΈ = sin A = π π cos ΞΈ = cos A = π π tan ΞΈ = tan A = π π csc ΞΈ = csc A = π π sec ΞΈ = sec A = π π cot ΞΈ = cot A = π You can use mnemonics! S oh C ah T oa S oh C ah T oa πππ. sin ΞΈ = βπ¦π. πππ. cos ΞΈ = βπ¦π. πππ. tan ΞΈ = πππ. You can use mnemonics! C ho S ha C ao C ho βπ¦π. csc ΞΈ = πππ. S ha βπ¦π. sec ΞΈ = πππ. C ao πππ. cot ΞΈ = πππ. ο It will be noted that the first three functions are the reciprocals of the other three, thus we may write: 1 sin ΞΈ = csc ΞΈ 1 cos ΞΈ = sec ΞΈ 1 tan ΞΈ = cot ΞΈ 1 csc ΞΈ = sin ΞΈ 1 sec ΞΈ = cos ΞΈ 1 cot ΞΈ = tan ΞΈ Note: The reciprocal of a number is 1 divided by the number. Pythagorean Theorem ο In a right triangle, to find the value of side c which is the hypotenuse of triangle ABC, we may use the Pythagorean theorem which states that: the square of the hypotenuse of a right triangle equals the sum of the squares of its legs. In symbols: c2 = a2 + b2 Derivations: c= a = ππ βππ ππ +ππ b = ππ βππ Examples 1. In a right triangle ABC where angle C is right, and sides a and b have lengths 3 and 4 units, respectively, find the measure of side c and derive the six trigonometric functions with reference to angle A. 12 2. Given sin A = , A is acute; find the 13 other trigonometric function values of A. Exercises 1. 2. 3. Draw the right triangle whose sides are a = 2, b = 4, and c = 20 and find the six trigonometric functions of angle B. Solve for the unknown side using the Pythagorean theorem given the sides a = 2 and b = 3. If tangent A = 5, what are the other trigonometric function values of A, given A is acute? Do Worksheet 3 Trigonometric Functions of Any Angle ΞΈ The trigonometric functions, which we are now ready to define, are illustrations of functions whose domains are sets of angles. ο Using the Cartesian Coordinate System, we will define the six trigonometric ratios. ο Let ΞΈ be an angle in standard position. Choose any point P(x, y) on the terminal side of ΞΈ. Draw a perpendicular line from P to the x-axis, thus forming a right triangle of reference for ΞΈ. ο ο The point P has three coordinates: the abscissa x, the ordinate y, and the hypotenuse r in terms of which we define the following trigonometric functions. sin ΞΈ = πππ πππππ ππππππππππ cos ΞΈ = ππππππππ ππππππππππ tan ΞΈ = πππ πππππ ππππππππ csc ΞΈ = ππππππππππ πππ πππππ sec ΞΈ = ππππππππππ ππππππππ cot ΞΈ = ππππππππ πππ πππππ π = π π = π π = π π = π π = π π = π If the coordinates of P(x, y) are given, we can derive the six trigonometric functions of angle ΞΈ in standard position. ο Since r is the hypotenuse of the right triangle, we can solve for its value using ο the Pythagorean relation: r = ππ + ππ Signs of the Six Trigonometric Functions Note: r is always positive. Thus, the signs of the values of the trigonometric functions of angle ΞΈ are determined by the signs of x and y. ο The signs of the trigonometric functions depend upon the quadrant in which the terminal side of ΞΈ lies. A S T C - All Students Take Calculus! A S T C - All Silly Trig Classes! Examples 1. 2. The terminal side of angle ΞΈ, in standard position, passes through the point P(β3, 4). Draw ΞΈ and find the values of the six trigonometric functions of ΞΈ. Let ΞΈ be an angle in standard position. Given 12 cos ΞΈ = . Find the other trigonometric 13 function values if the terminal side is in the fourth quadrant. Reference Angle The trigonometric function values of any angle ΞΈ can be determined by knowing the function values of acute angles (an angle Ο having positive radian measure less than or 2 less than 90°). The procedure involves the concept of a reference angle. ο Reference angle β is defined as the smallest positive acute angle ΞΈ associated with a nonquadrantal angle in standard position formed by the terminal side and the x-axis. ο Reference Angles ο To find the reference angle ΞΈβ of an angle ΞΈ (given angle) with positive rotation and whose measure is less than 360° or 2Ο, the following diagrams illustrate the relationship between ΞΈ and ΞΈβ in each of the four quadrants. In this case, ΞΈ = A. ο The following table summarizes how to find the reference angle if the given angle ΞΈ < 360°. If the given angle ΞΈ is in quadrant Reference angle ΞΈβ is (Degrees) (Radians) I II III ΞΈ 180° - ΞΈ ΞΈ - 180° ΞΈ Ο-ΞΈ ΞΈ-Ο IV 360° - ΞΈ 2Ο - ΞΈ ο To find the reference angle for an angle with negative measures or for an angle greater than 360°, first find a coterminal angle whose measure is between 0° and 360°. Then, use the appropriate formula summarized in the given table. Examples Given ΞΈ is an angle in standard position. Find the measure of its reference angles ΞΈβ. 1. 250° 2. 330° 3. 125° 45β Exercises I. In which quadrants do tan ΞΈ and sin ΞΈ have opposite signs? II. The terminal side of an angle Ξ± in standard position passes through the point M(-3, -3). Find the trigonometric functions of Ξ±. III. Given ΞΈ is an angle in standard position. Find the measure of its reference angle. 1. 2. 3. 4. 5Ο 6 23Ο 12 β120° 470° Do Worksheet 4