MAT 129 Precalculus Trigonometry Review Angles and their Measures - PowerPoint PPT Presentation

MAT 129 ‐ Trig MAT 129 – Precalculus Trigonometry Review Angles and their Measures David J. Gisch Angles Angles 1

MAT 129 ‐ Trig Angles and Quadrants Degrees and Radians • A radian is the size of an angle that subtends an arc, the length of the radius. • It turns out that the size of the radius does not matter. Radians Radians/ Degrees Recall that the circumference of a circle is 2�� . If the radius is 1, the circumference is 2� (i.e. the arc length of the whole circle is 2� ). 360° � 2� � 180° � 2� 2

MAT 129 ‐ Trig Radians/ Degrees Arc Length For a circle of radius � , a central angle of � radians subtends and arc whose length is � � �� Arc Length Arc Length Example 7.1.1: Find the length of the arc of a circle of radius 4 Example 7.1.2: Find the length of the arc of a circle of radius 10 meters subtended by a central angle of 30° . meters subtended by a central angle of 0.5 radian. 3

MAT 129 ‐ Trig Area of a S ector Arc Length Example 7.1.3: Find the area of the sector of a circle of radius 5 The area � of the sector of a circle with radius � formed feet formed by an angle of 40° . by a central angle of � radians is � � 1 2 � � � Angular/ Linear S peed S peed Example 7.1.4: A child is spinning a rock at the end of a 2-foot rope at the rate of 180 revolutions per minute (rpm). Find the linear speed of the rock when it is released. Linear Speed Angular Speed � � � � � 4

MAT 129 ‐ Trig S peed Example 7.1.5: The diameter of each wheel of a bicycle is 26 inches. If you are traveling at a speed of 35 miles per hour on this bicycle, how many revolutions per minute are the wheels turning. Right Triangle Trigonometry Pythagorean Theorem Right Triangle Trig 5

MAT 129 ‐ Trig Right Triangle Trig Right Triangle Trig Example 7.2.1: Find the value of each of the six trigonometric functions of the angle  . Right Triangle Trig Inverse Functions 6

MAT 129 ‐ Trig Right Triangle Trig Example 7.2.2: A person whose eyes are 5 feet above the ground is standing on the runway of an airport 100 feet from the control tower. That person observes an air traffic controller at the window of the 132-foot tower. What is the angle of elevation? Special Triangles 45-45-90 30-60-90 • If you have a right triangle with angles of 45° the sides • If you have a right triangle with angles of 30° and 60° the are always in the ratio of sides are always in the ratio of ℓ � ℓ � ℓ 2 ℓ � ℓ 2 � 2ℓ 7

MAT 129 ‐ Trig S pecial Triangles Right Triangle Trig Example 7.3.1: Find the exact value of each expression.         2 2 (a) sin tan (b) sin60 cos45 (c) cos cot 3 4 3 4 Recall S pecial Triangles The Unit Circle 8

MAT 129 ‐ Trig Unit Circle Unit Circle �1, 0� 1 30° �0, 1� Unit Circle THE Unit Circle 9

MAT 129 ‐ Trig Unit Circle and Trig Values Unit Circle and Trig Values 1 1 2 30° 3 2 Right Triangle Trig Example 7.4.1: Find the exact value of each expression. sin 30 cos 30 � sin�120� a) cos�225� sin�315� � 4 cos�60� b) �� �� 2 tan � 2 cos c) � � 10