MTH 133: Plane Trigonometry
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1 MTH 133: Plane Trigonometry The Trigonometric Functions Right Angle Trigonometry Thomas W. Judson Department of Mathematics & Statistics Stephen F. Austin State University Fall 2017 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12 Contents I 1 Course Orientation 2 Section 1.4. The Trigonometric Functions 3 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12
2 Course Orientation Overview Section 1.4. The Trigonometric Functions Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12 Course Orientation Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12
3 Section 1.4. The Trigonometric Functions The Trigonometric Functions Definition If (x, y) is any point on the origin of the terminal side of an angle α in the standard position on the unit circle x 2 + y 2 = 1, them sin α = y cos α = x tan α = y x csc α = 1 y sec α = 1 x cot α = x y, provided the denominator is not zero. Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12 Section 1.4. The Trigonometric Functions Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12
4 Section 1.4. The Trigonometric Functions The Reciprocal Identities Fact csc α = 1 sin α sec α = 1 cos α cot α = 1 tan α Find each of the following. 1 cos 45 2 tan 90 3 sin 30 4 sin 60 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12 Section 1.4. The Trigonometric Functions Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12
5 Section 1.4. The Trigonometric Functions Complete the Following Table α sin α cos α tan α csc α sec α cot α 0 π/6 π/4 π/3 π/2 2π/3 π 5π/4 7π/6 3π/2 5π/3 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12 Section 1.4. The Trigonometric Functions Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12
6 Inverse Trigonometric Functions Definition 1 sin 1 (x) = α provided sin α = x and π/2 α π/2 2 cos 1 (x) = α provided cos α = x and 0 α π 3 tan 1 (x) = α provided tan α = x and π/2 α π/2 Notation We sometimes write arcsin x instead of sin 1 x. It is important to remember that sin 1 does not mean 1 sin x. Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12
7 Inverse Trigonometric Functions Exercise Find each of the following: ) ( 1 1 cos 1 2 ( ) 2 2 sin tan 1 ( 3 ) 4 tan ( ( )) 3 sin 1 5 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12
8 Right Angle Trigonometry Fact If α is an acute angle in a right triangle, then sin α = opp hyp csc α = hyp opp cos α = adj hyp hyp sec α = adj tan α = opp adj cot α = adj opp, provided the denominator is not zero. Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12
9 Solving a Right Triangle Remarks To solve a right triangle: 1 Use the Pythagorean Theorem to find the length of the third side of the triangle when the length of the other two sides are know. 2 Use the fact that the sum of the angles of a triangle is 180 to find all of the angles of the triangle. 3 Use the trigonometric rations to find the missing sides and/or angles. Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12
10 Solving a Right Triangle Examples Example 1 Solve the right triangle in which α = 30 and c = 6. 2 Solve the right triangle in which a = 4 and b = 6. β c a α b γ = 90 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12
11 Modeling Problems Exercise A hiker stands 80 feet form a giant redwood tree and sights the top with an angle of elevation of 75. How tall is the tree (to the nearest foot)? Exercise Robin plans to use a 30 foot ladder to reach the castle window of Marion. Little John, who made the ladder, advises robin that the angle of elevation for the ladder must be between 55 and 70 for safety. What are the minimum and maximum heights that can be safely reached when the top of the ladder is placed against the castle wall (round to the nearest thenth)? Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12
12 Modeling Problems Exercise The sides of a regular pentagon are each 2 meters in length. Find the height h of the pentagon to the nearest hundredth of a meter. 2 m h Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall / 12
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