MATH 1204 UNIT 5: GEOMETRY AND TRIGONOMETRY Assumed Prior Knowledge Similar Right Triangles Recall that a Right Triangle is a triangle containing one 90 and two acute angles. Right triangles will be similar if an acute angle of one is equal to an acute angle of the other. Practical situations frequently occur in which similar right triangles are used to model and solve real-world problems EXAMPLE 1: A mast at the top of a building casts a shadow whose tip is 48 feet from the base of the building. If the building is 12 feet high and its shadow is 32 feet long, what is the length of the mast? (NOTE: If the length of the mast is x, then the height of the mast above the ground is x + 12.) Unit 5: Geometry and Trigonometry 1
The Pythagorean Theorem The Pythagorean Theorem states... "In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs." c 2 = a 2 + b 2 The Proof: On the following figure we have a right triangle with a square associated with each of its sides: Using the dimensions associated with the three sides, calculate the area of each of the squares. Then make sure that the area of the hypotenuse's square equals the areas of the other two squares together. Unit 5: Geometry and Trigonometry 2
EXAMPLE 2: On a baseball diamond the bases are 90 ft apart. What is the distance from home plate to second base in a straight line? Right Triangle Trigonometry Right Triangle Trigonometry is a mathematical means to indirectly measure unknown sides and angles of right triangles using trigonometric functions. Right Triangle Trigonometry is also known as Right Angle Trigonometry. We begin to study Right Triangle Trigonometry by naming the sides of the right triangle. Trigonometric Functions Trigonometric Ratios Measures Ratios Triangle Number Opposite to <A Adjacent to <A Hypotenuse Opposite Hypotenuse Adjacent Hypotenuse Opposite Adjacent 1 2 3 Observe the values you recorded in the table. What do you notice? Unit 5: Geometry and Trigonometry 3
The Three Basic Trigonometric Functions The 3 trigonometric ratios in the table above are the basis of the three main trigonometric functions: In the functions above, A denotes the angle in question, listed in degrees. How Do We Remember?? Trig Functions..... Soh-Cah Wha?!* We may use acronyms and other fun ways to memorize the three main trigonometric functions. For example, SOHCAHTOA (pronounced so-cah-toa) is one common acronym used to remember the functions. SOH Sin x = Opposite CAH Cos x = Adjacent TOA Tan x = Opposite Hypotenuse Hypotenuse Adjacent Unit 5: Geometry and Trigonometry 4
Here is another example of a way to memorize the trig functions: Some Old Hags Can't Acquire Husbands Til Old Age Group Activity: Try to come up with some acronyms of your own that will help you remember the three trigonometric functions! Using Trigonometric Functions: Example 3: Find the value of x in the diagrams below for the given information: Applications: Example 4: You are trying to climb onto the roof of your house which is 6 meters from the ground. If you have a ladder that is 12 meters long and you place it on the edge of the house at an angle of 50º from the ground, can you climb onto the roof of your house safely? Explain your answer. If you cannot climb onto your roof safely, suggest things that you could change so that you can climb onto your roof safely. Unit 5: Geometry and Trigonometry 5
HOMEWORK ASSIGNMENT 1. From the triangle below state the ratios: (o-opposite, a-adjacent, h-hypotenuse) a) o/h = b) a/h = c) o/a = 2. Suppose you have been assigned to measure the height of the local water tower. Climbing makes you dizzy, so you decide to do the whole job at ground level. From a point 47.3 meters from the base of the water tower, you find that you must look up at an angle of 53 to see the top of the tower. How tall is the tower? Draw the triangle. 3. Scientists estimate the heights of features on the moon by measuring the lengths of the shadows they cast on the moon s surface. From a photograph, you find that the shadow cast on the inside of a crater by its rim is 325 meters long. At the time the photograph was taken, the sun s angle to the horizontal surface was 23.6. How high does the rim rise above the inside of the crater? Unit 5: Geometry and Trigonometry 6