Geometry Day 60 Trigonometry II: Sohcahtoa’s Revenge

Today’s Objective More with Trigonometry Applications Angles of Elevation and Depression Trigonometry and the Cartesian Plane Vectors

Review The trigonometric functions – sine, cosine, and tangent – give the ratios of certain sides given an acute angle of a right triangle. Remember, you take the sin, cos, and tan of angles, and you will get a ratio. The inverses of sin, cos, and tan will take the ratio and give you the angle.

Applications There is a cell phone tower with a guide wire going from the ground to the top of the tower. The tower is 100 ft tall and the wire is connected to the ground at a 70  angle. How long is the wire? Draw a diagram and use trigonometry to calculate the answer.

Elevation and Depression An angle of elevation is the angle above horizontal one must look up at to sight an object. An angle of depression is the angle below horizontal one must look down at to sight an object. These are always formed with horizontal lines, never vertical!! Note that the angles of depression and elevation are the same. Why?

Applications A hunter is in a tree, 12 meters off the ground. He sees a lion 50 meters away from the base of the tree. What is the angle of depression he would have to aim with in order to shoot the lion? Draw a diagram and use trigonometry to calculate the answer.

Indirect Measurement We can use angles of elevation or depression to measure the distance between two objects, or we can use two different angles of elevation/depression of a single object to estimate its height. Ms. Hood wishes to have a tree cut down. To calculate its height, she sights the top of the tree at a 70  angle of elevation. She then moves back 10 meters and sights the treetop at 26 . If Ms. Hood’s eye-height is 1.6 meters, how tall is the tree? Draw a diagram and calculate the answer.

Indirect Measurement Another example: Two buildings sighted from the top of a 200 ft building. The base of the first building is sighted at an angle of depression of 35  ; the base of the second is sighted an angle of depression of 36 . How far apart are the two buildings? Draw a diagram and calculate the answer.

Tangent and the Cartesian Plane Take out a sheet of graph paper. Graph the following line: What is the angle of elevation of this line? Discuss with your neighbors how you can use what we’ve talked about to solve this problem.

Tangent and the Cartesian Plane The angle of elevation of a line can be found by drawing a right triangle, using the line as the hypotenuse, and using inverse tangent to get the angle. Is there a relationship between the equation of the line, and the sides of the triangle you draw? The slope of the line represent the opposite (rise) and adjacent (run) legs. So when you find the inverse tangent, you are using the same numbers that make up the slope.

Vectors A vector is a quantity that has both magnitude (size) and direction. Vectors are named after their endpoints (two uppercase letters) or a single lowercase letter specific to a given vector. Ex: or See p. 593 for diagram The magnitude of a vector is the length of the vector from its initial point to its terminal point. Magnitude is noted with vertical bars: The direction of a vector is the angle that is formed with the positive x-axis (or any horizontal line).

Vectors Vectors are graphed as directed line segments (with an arrow on one end). Vector has an initial point A and terminal point B. When on the coordinate plain, a vector is in standard position when it has its initial point at the origin (0, 0). The component form of a vector describes its horizontal change x and vertical change y from its initial point to its terminal point. The component form of a vector is written with pointy brackets: Example: write the component form of a vector with initial point (5, -1) and terminal point (-2, 3).

Finding Magnitude and Direction You can find the magnitude of a vector with the Distance Formula/Pythagorean Theorem. You can use trigonometry to find the direction of a vector: Find the absolute value of the slope of the vector. Take the inverse tangent of this value. This gives you the angle the vector makes with the x-axis (when it’s repositioned into standard form), but it is very important to note what quadrant the terminal point is in. Example: find the magnitude and direction of with R(2, 4) and T(-3, -2).

Vector Types See examples and non-examples on p. 594 Two vectors are equal if and only if they have the same magnitude and direction. Two vectors are parallel if and only if they have the same or opposite direction. Two vectors are opposite if and only if they have the same magnitude and opposite directions.

Vector Addition The sum of two vectors is the resultant. There are two methods for vector addition (p. 595): Parallelogram method Triangle method Subtracting a vector is the same as adding its opposite. Vectors can be added or subtracted algebraically by adding or subtracting their horizontal and vertical components.

Assignments Homework 35 Workbook, p. 106 Handout (Linear equations & Tangent), Homework 36 Workbook, pp. 109, 110 (#9 only)