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1 Trigonometric Functions Copyright © 2009 Pearson Addison-Wesley 1.2-1 1 Trigonometric Functions 1.1 Angles 1.2 Angle Relationships and Similar Triangles 1.3 Trigonometric Functions 1.4 Using the Definitions of the Trigonometric Functions Copyright © 2009 Pearson Addison-Wesley 1.2-2 1.2 Angle Relationships and Similar Triangles Geometric Properties ▪ Triangles Copyright © 2009 Pearson Addison-Wesley 1.1-3 1.2-3 Vertical Angles Vertical angles have equal measures. The pair of angles NMP and RMQ are vertical angles. Copyright © 2009 Pearson Addison-Wesley 1.1-4 1.2-4 Parallel Lines Parallel lines are lines that lie in the same plane and do not intersect. When a line q intersects two parallel lines, q, is called a transversal. Copyright © 2009 Pearson Addison-Wesley 1.2-5 Angles and Relationships Name Angles Rule Alternate interior angles 4 and 5 3 and 6 Angles measures are equal. Alternate exterior angles 1 and 8 2 and 7 Angle measures are equal. Interior angles on the same side of the transversal 4 and 6 3 and 5 Angle measures add to 180. Corresponding angles 2 & 6, 1 & 5, 3 & 7, 4 & 8 Angle measures are equal. Copyright © 2009 Pearson Addison-Wesley 1.2-6 Example 1 FINDING ANGLE MEASURES Find the measure of angles 1, 2, 3, and 4, given that lines m and n are parallel. Angles 1 and 4 are alternate exterior angles, so they are equal. Subtract 3x. Add 40. Divide by 2. Angle 1 has measure Copyright © 2009 Pearson Addison-Wesley 1.1-7 Substitute 21 for x. 1.2-7 Example 1 FINDING ANGLE MEASURES (continued) Angle 4 has measure Substitute 21 for x. Angle 2 is the supplement of a 65° angle, so it has measure . Angle 3 is a vertical angle to angle 1, so its measure is 65°. Copyright © 2009 Pearson Addison-Wesley 1.1-8 1.2-8 Angle Sum of a Triangle The sum of the measures of the angles of any triangle is 180°. Copyright © 2009 Pearson Addison-Wesley 1.1-9 1.2-9 Example 2 APPLYING THE ANGLE SUM OF A TRIANGLE PROPERTY The measures of two of the angles of a triangle are 48 and 61. Find the measure of the third angle, x. The sum of the angles is 180°. Add. Subtract 109°. The third angle of the triangle measures 71°. Copyright © 2009 Pearson Addison-Wesley 1.1-10 1.2-10 Types of Triangles: Angles Copyright © 2009 Pearson Addison-Wesley 1.2-11 Types of Triangles: Sides Copyright © 2009 Pearson Addison-Wesley 1.2-12 Conditions for Similar Triangles For triangle ABC to be similar to triangle DEF, the following conditions must hold. 1. Corresponding angles must have the same measure. 2. Corresponding sides must be proportional. (That is, the ratios of the corresponding sides must be equal.) Copyright © 2009 Pearson Addison-Wesley 1.1-13 1.2-13 Example 3 FINDING ANGLE MEASURES IN SIMILAR TRIANGLES In the figure, triangles ABC and NMP are similar. Find the measures of angles B and C. Since the triangles are similar, corresponding angles have the same measure. B corresponds to M, so angle B measures 31°. C corresponds to P, so angle C measures 104°. Copyright © 2009 Pearson Addison-Wesley 1.1-14 1.2-14 Example 4 FINDING SIDE LENGTHS IN SIMILAR TRIANGLES In the figure, triangles ABC and NMP are similar. Find the measures of angles B and C. Since the triangles are similar, corresponding sides are proportional. DF corresponds to AB, and DE corresponds to AC, so Copyright © 2009 Pearson Addison-Wesley 1.1-15 1.2-15 Example 4 FINDING SIDE LENGTHS IN SIMILAR TRIANGLES (continued) Side DF has length 12. EF corresponds to CB, so Side EF has length 16. Copyright © 2009 Pearson Addison-Wesley 1.1-16 1.2-16 Example 5 FINDING THE HEIGHT OF A FLAGPOLE Firefighters at a station need to measure the height of the station flagpole. They find that at the instant when the shadow of the station is 18 m long, the shadow of the flagpole is 99 ft long. The station is 10 m high. Find the height of the flagpole. Since the two triangles are similar, corresponding sides are proportional. Copyright © 2009 Pearson Addison-Wesley 1.1-17 1.2-17 FINDING THE HEIGHT OF A FLAGPOLE (continued) Example 5 Lowest terms The flagpole is 55 feet high. Copyright © 2009 Pearson Addison-Wesley 1.1-18 1.2-18