1.1 Angles Basic Terminology ▪ Degree Measure ▪ Standard Position ▪ Coterminal Angles Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

1.1 Example 1 Finding the Complement and the Supplement of an Angle (page 3) For an angle measuring 55°, find the measure of its complement and its supplement. Complement: 90° − 55° = 35° Supplement: 180° − 55° = 125° Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Find the measure of each angle. 1.1 Example 2(a) Finding Measures of Complementary and Supplementary Angles (page 3) Find the measure of each angle. The two angles form a right angle, so they are complements. The measures of the two angles are and Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Find the measure of each angle. 1.1 Example 2(b) Finding Measures of Complementary and Supplementary Angles (page 3) Find the measure of each angle. The two angles form a straight angle, so they are supplements. The measures of the two angles are and Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

1.1 Example 3 Calculating with Degrees, Minutes, and Seconds (page 4) Perform each calculation. (a) (b) 180˚ 0′ –117˚29′ 179˚60′ –117˚29′ Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

(a) Convert 105°20′32″ to decimal degrees. 1.1 Example 4 Converting Between Decimal Degrees and Degrees, Minutes, and Seconds (page 5) (a) Convert 105°20′32″ to decimal degrees. (b) Convert 85.263° to degrees, minutes, and seconds. Show them how to use the M+ and Mc on basic Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

1.1 Example 5 Finding Measures of Coterminal Angles (page 6) Find the angles of least possible positive measure coterminal with each angle. Add or subtract 360° as many times as needed to obtain an angle with measure greater than 0° but less than 360°. (a) 1106° An angle of 1106° is coterminal with an angle of 26°. (b) –150° An angle of –150° is coterminal with an angle of 210°. (c) –603° An angle of –603° is coterminal with an angle of 117°.

1.1 Example 6 Analyzing the Revolutions of a CD Player (page 7) A wheel makes 270 revolutions per minute. Through how many degrees will a point on the edge of the wheel move in 5 sec? The wheel makes 270 revolutions in one minute or revolutions per second. In five seconds, the wheel makes revolutions. Each revolution is 360°, so a point on the edge of the wheel will move Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

1.1 Summary Supplementary: two angles with a sum of 180˚ Complementary: two 2 angles with a sum of 90˚ Convert from decimal degrees to degrees, minutes, & seconds Convert from degrees, minutes, & seconds to decimal degrees Coterminal angles ± 360˚ Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

1.2 Angles Geometric Properties ▪ Triangles Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

1.2 Example 1 Finding Angle Measures (page 12) Find the measures of angles 1, 2, 3, and 4 in the figure, given that lines m and n are parallel. Angles 2 and 3 are same side interior, so they are supplements.

1.2 Example 5 Finding the Height of a Tree (page 15) Samir wants to know the height of a tree in a park near his home. The tree casts a 38-ft shadow at the same time as Samir, who is 63 in. tall, casts a 42-in. shadow. Find the height of the tree. Let x = the height of the tree The tree is 57 feet tall. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

1.2 summary Geometric properties and Similar Triangles Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Trigonometric Functions 1.3 Trigonometric Functions Trigonometric Functions ▪ Quadrantal Angles Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Trigonometry Rules Need to memorize: Soh Cah Toa Sin θ = y/r r Cos θ = x/r y θ Tan θ = y/x x Csc θ = r/y Sec θ = r/x Cot θ = x/y Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

1.3 Example 1 Finding Function Values of an Angle (page 23) The terminal side of an angle θ in standard position passes through the point (12, 5). Find the values of the six trigonometric functions of angle θ. x = 12 and y = 5. 13 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

1.3 Example 2 Finding Function Values of an Angle (page 23) The terminal side of an angle θ in standard position passes through the point (8, –6). Find the values of the six trigonometric functions of angle θ. x = 8 and y = –6. 10 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

1.3 Example 3 Finding Function Values of an Angle (page 25) Find the values of the six trigonometric functions of angle θ in standard position, if the terminal side of θ is defined by 3x – 2y = 0, x ≤ 0. Since x ≤ 0, the graph of the line 3x – 2y = 0 is shown to the left of the y-axis. Find a point on the line: Let x = –2. Then A point on the line is (–2, –3). Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

1.3 Example 3 Finding Function Values of an Angle (cont.) Point (-2,-3) -2 -3 √13 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Find the values of the six trigonometric functions of a 360° angle. 1.3 Example 4(a) Finding Function Values of Quadrantal Angles (page 26) Find the values of the six trigonometric functions of a 360° angle. The terminal side passes through (2, 0). So x = 2 and y = 0 and r = 2. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

1.3 Example 4(b) Finding Function Values of Quadrantal Angles (page 26) Find the values of the six trigonometric functions of an angle θ in standard position with terminal side through (0, –5). x = 0 and y = –5 and r = 5. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

1.3 Summary Trigonometry Rules Need to memorize: Soh Cah Toa Sin θ = y/r r Cos θ = x/r y θ Tan θ = y/x x Csc θ = r/y Sec θ = r/x Cot θ = x/y Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Using the Definitions of the Trigonometric Functions 1.4 Using the Definitions of the Trigonometric Functions Reciprocal Identities ▪ Signs and Ranges of Function Values ▪ Pythagorean Identities ▪ Quotient Identities Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

1.4 Example 1 Using the Reciprocal Identities (page 31) Find each function value. (a) tan θ, given that cot θ = 4. tan θ is the reciprocal of cot θ. (b) sec θ, given that sec θ is the reciprocal of cos θ. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

All Students Take Calculus Need to memorize: Students + sin - cos & tan All + sin, cos, & tan + csc - sec & cot Take + tan - cos & sin Calculus + cos - sin & tan + cot - sec & csc + sec - csc & cot Which others are also positive and negative? (the reciprocals) Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

1.4 Example 2 Finding Function Values of an Angle (page 32) Determine the signs of the trigonometric functions of an angle in standard position with the given measure. (a) 54° (b) 260° (c) –60° Students Students All + sin, cos, & tan + csc, sec, cot All Students All Take Calculus Take + tan, cot - sin, cos, sec, csc Calculus Take Calculus + cos, sec - sin, tan, csc, cot Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

1.4 Example 3 Identifying the Quadrant of an Angle (page 33) Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. (a) tan θ > 0, csc θ < 0 Where is the tan positive and csc negative Where is the tan positive and sin negative Answer: Quadrant III II I Students All Take Calculus (b) sin θ > 0, csc θ > 0 III IV Where is the sin positive and csc positive Where is the sin positive and sin positive Answer: Quadrant I & II Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Need to Memorize Tan and Cot range = (-∞,∞) Sin and Cos range = [-1,1] Csc and Sec range = (-∞,-1] [1,∞) ∩ Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Decide whether each statement is possible or impossible. 1.4 Example 4 Deciding Whether a Value is in the Range of a Trigonometric Function (page 33) Use the picture to show why Decide whether each statement is possible or impossible. (a) cot θ = –.999 (b) cos θ = –1.7 (c) csc θ = 0 (a) cot θ = –.999 is possible because the range of cot θ is cot θ = -.999 tan θ = -1/.999 θ = tan-1 (-1/.999) (b) cos θ = –1.7 is impossible because the range of cos θ is [–1, 1]. cos θ = -1.7 θ = cos-1 (-1.7) = error (c) csc θ = 0 is impossible because the range of csc θ is csc θ = 0 sin θ = 1/0 1/0 undefined

1.4 Example 5 Finding All Function Values Given One Value and the Quadrant (page 33) Angle θ lies in quadrant III, and Find the values of the other five trigonometric functions. Since θ lies in quadrant III, then x is negative and y is negative x=-5 θ y=-8 r= Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Find cos θ and tan θ given that sin θ and cos θ > 0. 1.4 Example 6 Finding Other Function Values Given One Value and the Quadrant (page 35) Find cos θ and tan θ given that sin θ and cos θ > 0. Find which quadrant first? Where is the sin negative and cos positive? x= θ y= r=3 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Find cot θ and csc θ given that cos θ and θ is in quadrant II. 1.4 Example 7 Finding Other Function Values Given One Value and the Quadrant (page 36) Find cot θ and csc θ given that cos θ and θ is in quadrant II. Since θ is in quadrant II, x is negative and y is positive r=25 y=24 θ x=-7 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

Additional Example Given csc θ = 5.3467892 and θ is in quadrant II, find cot θ Since csc θ = 5.3467892, then sin θ = 1/5.3467892 r=5.3467892 x = -√(5.34678922 – 12) = -5.25244741 y=1 θ x= Need to find the cot θ which is x/y cot θ = -5.25244741 Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

1.4 Summary Trigonometry Rules Need to memorize: Soh Cah Toa Students All Sin θ = y/r r Cos θ = x/r y θ Tan θ = y/x x Csc θ = r/y Take Calculus Sec θ = r/x Cot θ = x/y Copyright © 2008 Pearson Addison-Wesley. All rights reserved.