ALGEBRA II Chapter 13 Notes The word trigonometry is derived from the ancient Greek language and means measurement of triangles. Section 13.1 Right-Triangle Trigonometry Objectives: 1. Find the trigonometric functions of acute angles. 2. Solve a right triangle by using trig. functions. Let s review a right triangle. It has 3 sides: 2 legs and a hypotenuse, and three angles: two acute and one right angle. The angles are usually in capital letters with their opposite side in small letters. You only use acute angles with the trigonometric functions. 1. Label the sides (a, b, c) 2. Label the sides. A (opp, adj, hyp) 3. Label the sides. B A trigonometric ratio is a, The six trigonometric functions are defined below with these abbreviations. 4. Write the ratio of the sides with their letter names. Trigonometric Functions of A sin A opp. hyp. cos A adj. hyp. hyp. csc A opp. hyp. sec A adj. tan A opp. adj. adj. cot A opp. The three basic trig functions are the sine, cosine and the tangent. This saying helps us remember. SOHCAHTOA Example 1. Find the values of the six trigonometric functions of X for XYZ at right. Give the exact and answers rounded to the nearest ten-thousandth. The cosecant, secant, and cotangent ratios can be expressed in terms of sine, cosine, and tan ratios, respectively. csc 1 sin, sec 1 cos, cot 1 tan.
Example 2 Refer to FGH to find each value listed. Give exact answers and answers rounded to the nearest ten thousandth (4 decimal places). 1. sin G = 7. sin F = 2. cos G = 8. cos F = 3. tan G = 9. tan F = 4. csc G = 10. csc F = 5. sec G = 11. Sec F = 6. cot G = 12. cot F = Example 3 A tipping platform is a ramp used to unload trucks. How high is the end of an 80 inch ramp when it is tipped by a 30º angle? By a 45º angle? When you know the trigonometric ratio of an angle you can find the measure of that angle by using the inverse relation of the trigonometric. For example if tan A is ¾ then the angle whose tangent is ¾ is written tan 1 ¾. Thus the m A tan 1 ¾. m A 37 Note: Most calculators have,, keys for these inverse trigonometric relations. Note that is not the same as. sin 1 the angle measure. Example 4 Find the m A by using inverse trigonometric functions. Solving a triangle involves finding the measures of all of the unknown sides and angles of the triangle. Helpful hints: The sum of the two acute angles is 90. If you know two sides of the right triangle, use the Pythagorean Theorem to find the third side. Example 5 Solve the right triangles.
Section 13.2 Angles of Rotation Objectives: 1. Find conterminal and reference angles 2. Find the trig function values of angles in standard form. In geometry, an angle is defined by two rays that have the same common endpoint. In trigonometry, an angle is defined by a ray that is rotated around its endpoint. Each position of the rotated ray, relative to its stating position, creates an angle of rotation. The Greek letter Theta,, is commonly used to name an angle of rotation. The initial position of the ray is called the initial side of the angle, and the final position is called the terminal side of the angle. When the initial side lies along the positive x- axis and its endpoint is at the origin, the angle is said to be in standard position. If the direction of rotation is counterclockwise, the angle has a positive measure. If the direction if rotation is clockwise, the angle has a negative measure. The most common unit for angle measure is the degree. A complete rotation of a ray is assigned a measure of 360. Example 1 Sketch each angle in standard position a. 135º b. 270º c. -25º Angles in standard position are coterminal if they have the same terminal side. Example 2 Find the coterminal angle,, for each angle below such that 360 360. Add or subtract 360º from each angle. Discard answers that are not in the given range. a. 100º b. -295º c. 540º Reference angle: For an angle in standard position, the reference angle, ref, is the positive acute angle formed by the terminal side of and the nearest part (positive or negative) of the x-axis. Use the positive x-axis for angles in Quadrants I and IV and use the negative x-axis for angles in Quadrants II and III. Example 3 Find the reference angle, ref, for each angle. a. 94 b. 249 c. 290 d. 110 Try this: find the reference angle, ref, for 315 and 235.
If you think of x and y as the coordinates of a point on the terminal side of an angle in standard position, you will be able to determine the correct sign of the values for the trigonometric function. Example 4 Find the exact values of the six trigonometric functions of given each point in the terminal side of in standard position. a. (4, -3) b. (-8, 3) Example 5 Given the quadrant of in standard position and a trigonometric value of, find the exact values for the indicated functions. a. II, cos 3 ; tan b. II, sin 0.4; sec 5 Example 6 Find the number of rotations of the fraction of a rotation represented by each angle below. Indicate whether the rotation is clockwise or counterclockwise. a. -270º b. 640º
Section 13.3 Trigonometric Functions of any angle Objectives: Find exact values for trigonometric functions, of special angles and their multiples. Find approximate values for trigonometric functions of any angle. There are certain angles whose exact trigonometric function values can be found without a calculator. From Geometry, we learned that the lengths of the sides of a 45-45-90 triangle have a ratio of 1 to 1 to 2, and the lengths of the sides of a 30-60-90 triangle have a ratio of 1 to 3 to 2. You can use these relationships to find the exact values of the trigonometric angles. Example 1 Make a table of the exact values of the sine, cosine, and tangent of 30º, 45º, and 60º. sin cos tan 30º 45º 60º When a circle centered at the origin has a radius of 1 it is called a unit circle. Because the radius (r) is 1, the coordinates of P are (cos, sin ). cos is the x-value of the point, and sin is the y- value.
Example2 Find the exact values of the sine, cosine, and tangent of each angle. 1. 240º 2. -240º 3. 1305º 4. -840º Example 3 Find each trigonometric function value. Give exact answers. 1. cos120 2. sin( 90 ) 3. tan390
Section 13.4 Radian Measure and Arc Length Objectives: Convert from degree measure to radian measure and vice verse. Find arc length. A useful angle measure other than degrees is radian measure. The circumference of a circle is 2πr. Because the radius of a unit circle is 1, its circumference is 2π. An angle of rotation of 360º has a measure of 2π radians, an angle of 180º has a measure of π radians. You can convert from degrees to radians, and vise versa, by using the relationship below. Example 1 Convert each degree measure to radian measure. Give exact answers. a. 90º b. -120º c. 20º d. 400º e. 1080º f. 50º Example 2 Convert each radian measure to degree measure. Round answers to the nearest tenth of a degree, if necessary. a. 2 3 radians b. 9 radians c. 3 radians d. 3.235 radians Example 3 Evaluate. Give exact values. (Convert from radians to degrees. Then evaluate). a. sin b. cos 3 c. tan d. cos 7 3 2 4 4 Arc length: If is the radian measure of a central angle in a circle with a radius of r, then the length, s, of the arc intercepted by is s r. Example 3 A circle has a diameter of 10 meters. For each central angle measure below, find the length in meters of the arc intercepted by the angle. a. 4.28 radians b. 2 3
Section 13.5 Graphing Trigonometric Functions Objectives: 1. Graph sine cosine and tangent functions 2. Graphing the above functions with transformations Example 1 Graph the following parent graphs \a) y sin b) y cos c) y tan There are four parts you must know to graphing trigonometric functions, amplitude, phase shift, period, and vertical shift. Each are found by following the formulas to the right. Example 2 State the amplitude, phase shift, period, and vertical shift of each function then graph the function. a) y sin( 45 ) b) y cos 1 amplitude: period: amplitude: period: phase shift: vertical shift: phase shift: vertical shift: c) y 2sin b) y cos(2 ) amplitude: period: amplitude: period: phase shift: vertical shift: phase shift: vertical shift: c) y sin( 1 2 45 ) 1 b) y 2cos( 2 ) 1 2 amplitude: period: amplitude: period: phase shift: vertical shift: phase shift: vertical shift: