θ. The angle is denoted in two ways: angle θ

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1 1.1 Angles, Degrees and Special Triangles (1 of 24) 1.1 Angles, Degrees and Special Triangles Definitions An angle is formed by two rays with the same end point. The common endpoint is called the vertex of the angle, and the rays are called the sides of the angle. In the figure, the vertex of angle θ (theta) is labled O, and A and B are points on each side of θ. The angle is denoted in two ways: angle θ and angle AOB. Think of θ as being formed by rotating ray OA about its vertex to side OB. In this case, OA is called the initial side of θ and OB is called the terminal side of θ. When the rotation from the initial to the terminal side is counterclockwise, the the angle is positive, otherwise the angle is negative. An angle formed by rotating the rotating a ray one complete revolution has a measure of

2 1.1 Angles, Degrees and Special Triangles (2 of 24) Common Angles A right angle has measure 90 0, a straight angle is 180 0, an acute angle measures between 0 0 and 90 0, and an obtuse angle is over Two angles that are complementary have a sum of 90 0, and two angles that are supplementary have a sum of Example 1 Give the complement and supplement of each angle. a b c. θ

3 1.1 Angles, Degrees and Special Triangles (3 of 24) Triangles A triangle is a three sided polygon. It is customary to label the triangle so that side a opposite angle A, side b opposite angle B and side c opposite angle C. In an equalteral triangle all three sides are of equal length and all three angles are equal. An isoseles triangle has two equal sides and two equal angles. A scalene triangle no equal sides and no equal angles. An acute triangle has three acute angles. An obtuse triangle has exactly one obtuese angle. A right triangle has one right angle.

4 1.1 Angles, Degrees and Special Triangles (4 of 24) Pythagorean Theorem In any right triangle, the square of the length of the hypoteneuse (longest side) is equal to the sum of the squares of the legs (the other two sides). Example 2 Solve for x is the triangle shown.

5 1.1 Angles, Degrees and Special Triangles (5 of 24) Example 3 One of the chair lifts at a Tahoe ski resort has a vertical rise of 1,170 feet. If the length of the chair lift is 5,750 feet, find the horizontal distance covered by a person on the lift (round to the nearest foot). The Triangle In any right triangle that also has a 30 0 angle and a 60 0 angle, the hypotenuse is always twice the shortest side, and the middle-length side is always the shortest side times 3. Proof:

6 1.1 Angles, Degrees and Special Triangles (6 of 24) Example 4 If the shortest side of a triangle is 5, find the other two sides. Express the answers in exact form and then round to the nearest two decimal places. Example 5 A ladder is leaning against a wall. The top of the ladder is 4 feet above the ground and the bottom of the ladder make an angle of 60 0 with the ground. Express the answers in exact form and then round to the nearest two decimal places. a. How long is the ladder? b. How far from the wall is the bottom of the ladder?

7 1.1 Angles, Degrees and Special Triangles (7 of 24) The Triangle In any right triangle that also two 45 0 angles, then the legs have equal length and the hypotenuse is 2 times the length leg. Example 6 A 10-foot rope connects the top of a tent pole to the ground. If the rope makes an angle of 45 0 with the ground, find the length of the tent pole.

8 1.2 The Rectangular Coordinate System (8 of 24) 1.2 The Rectangular Coordinate System Example 1 y Graph y = 3 2 x x -2

9 1.2 The Rectangular Coordinate System (9 of 24) Vertex Form of a Parabola Any parabola can be described by the equation y = a(x h) 2 + k. The vertex of the parabola is at (h, k). If a > 0, then the parabola opens upward, and if a < 0, the parabola opens upward. The value of a also determines how wide or narrow the parabola is. Example 2 At a county fair a human cannonball is show from a canon. He reached a height of 70 feet before landing in a net 160 feet from the canon. Sketch the path of the human cannonball and find its equation.

10 1.2 The Rectangular Coordinate System (10 of 24) The Distance Formula The distance r between any two points (x 1, y 1 ), (x 2, y 2 ) in the rectangular coordinate system is given by. r = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 Example 3 Find the distance between the points ( 1, 5) and (2, 1).

11 Example 4 Find the distance between the point (x, y) and the origin. 1.2 The Rectangular Coordinate System (11 of 24)

12 1.2 The Rectangular Coordinate System (12 of 24) The Equation of a Circle A circle is the set of all points in the plane that are a fixed distance from a given fixed point. The distance is the radius of the circle r, and the fixed point is the center of the circle (h, k).. (x h) 2 + (y k) 2 = r 2 Example 5 Verify the points (1/ 2, 1/ 2) and ( 3 / 2, 1/ 2) both lie on the circle of radius 1 centered at the origin. This circle is called the unit circle.

13 1.2 The Rectangular Coordinate System (13 of 24) Angle in Standard Position An angle is in standard position if its initial side is along the positive x-axis and its vertex is at the origin. Example 6 Draw angle θ = 45 0 in standard position and find three points on the terminal side θ. Notation: When the terminal side of θ = 45 0 is in quadrant I, we denote it θ QI, read angle theta is an element of quadrant I. Quadrantal and Coterminal Angles If the terminal side of an angle lies on one of the axes (90 0, 180 0, 270 0, 360 0, etc...) the angle is called a quadrantal angle. Two angles in standard position with the same terminal side are coterminal angles.

14 1.2 The Rectangular Coordinate System (14 of 24) Example 7 Draw 90 0 in standard position and find two positive angles and two negative angles that are coterminal to Example 8 Find all angles coterminal to Common Angles in Trigonometry (Unit Circle)

15 1.3 Definition of the Six Trigonometric Functions (15 of 24) 1.3 Definition of the Trigonometric Functions

16 1.3 Definition of the Six Trigonometric Functions (16 of 24) Example 1 Find the six trigonometric functions of θ if θ is in standard position and the point (-2, 3) is on the terminal side of θ. Example 2 Find the sine and cosine of θ = Example 3 Find the six trigonometric functions of θ =

17 1.3 Definition of the Six Trigonometric Functions (17 of 24) Example 4 What is greater, tan 30 0 or tan How large can tanθ be? Signs of Trigonometric Functions Example 5 If sinθ = 5 /13 and θ terminates in quadrant III, find cosθ and tanθ.

18 1.4 Introduction to Identities (18 of 24) 1.4 Introduction to Identities Memorize The Six Trigonometric Functions Memorize The Reciprocal Identities

19 1.4 Introduction to Identities (19 of 24) Memorize The Reciprocal Identities Examples If sinθ = 3, find cscθ If cosθ = 3 2, find secθ. 3. If tanθ = 2, find cotθ. 4. If cscθ = a, find sinθ. 5. If secθ = 1, find cosθ. 6. If cotθ = 1, find tanθ.

20 1.4 Introduction to Identities (20 of 24) Memorize The Ratio Identities Examples 7 If sinθ = 3 5 and cosθ = 4, find tanθ and cotθ. 5 Notational Point: sin 2 θ = ( sinθ ) 2 Examples If sinθ = 3 5, find sin2 θ. 9. If cosθ = 1 2, find cos2 θ

21 Memorize the Pythagorean Identities 1.4 Introduction to Identities (21 of 24)

22 1.4 Introduction to Identities (22 of 24) Example 10 If sinθ = 3 / 5 and θ terminates in quadrant II, find cosθ and tanθ. Example 11 If cosθ = 1/ 2 and θ terminates in quadrant IV, find the remaining trigonometric functions evaluated at θ.

23 1.5 More on Identities (23 of 24) 1.5 More on Identities Example 1 Write tanθ in terms of sinθ [and no other trigonometric functions]. Example 2 Write secθ tanθ in terms of sinθ and cosθ. Example 3 Add 1 sinθ + 1 cosθ.

24 1.5 More on Identities (24 of 24) Example 4 Multiply (sinθ + 2)(sinθ 5). Example 5 Substitute x = 3tanθ in the expression x and simplify. Example 6 Prove the following identity. That is, show the following statement is true by transforming the left side to the right side. cosθ tanθ = sinθ Example 7 Prove the identity (sinθ + cosθ) 2 = 1+ 2sinθ cosθ