Hertel/Williams/Lambert Trigonometry packet Geometry honors Table of contents: Day 1: Basic Trigonometry Review SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles. Day 2: Trig Review and Co-Functions SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction. Day 3: Using Trigonometry to Determine Area SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known. Day 4: Law of Sines SWBAT: Find the missing side lengths of an acute triangle given one side length and the measures of two angles. Day 5: Law of Cosines SWBAT: Find the missing side lengths of an acute triangle given two side lengths and the measure of the included angle. Day 6: Review Day 7: TEST SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles. Day 1 Basic Trigonometry Review Warm Up: Review the basic Trig Rules below and complete the example below: Basic Trigonometry Rules: These formulas ONLY work in a right triangle. The hypotenuse is across from the right angle. Questions usually ask for an answer to the nearest units. You need a scientific or graphing calculator. Example: SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles. Example 1: Practice: 1.) SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles. 2.) Example 2: Example 3: Inverse Trigonometric Relations to find Missing Angles. SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles. Practice: 3.) 4.) 5.) SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles. Exit Ticket: Summary: SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles. Day 1 Homework: SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles. SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles. 8. 9. 10. 11. SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction. Day 2 Basic Trig Word Problems and CoFunctions Warm Up: 1.) Find the length of side p and the measure of angle m, as shown on the diagram. Give each answer correct to the nearest whole number or degree. 2.) SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction. Real World Connection: Trigonometry can be used on a daily basis in the workplace. Since trigonometry means "triangle measure", any profession that deals with measurement deals with trigonometry as well. Carpenters, construction workers and engineers, for example, must possess a thorough understanding of trigonometry. Angle of Elevation The angle of elevation is always measured from the ground up. Think of it like an elevator that only goes up. It is always INSIDE the triangle. In the diagram at the left, x marks the angle of elevation of the top of the tree as seen from a point on the ground. You can think of the angle of elevation in relation to the movement of your eyes. You are looking straight ahead and you must raise (elevate) your eyes to see the top of the tree. Angle of Depression The angle of depression is always OUTSIDE the triangle. It is never inside the triangle. In the diagram at the left, x marks the angle of depression of a boat at sea from the top of a lighthouse. You can think of the angle of depression in relation to the movement of your eyes. You are standing at the top of the lighthouse and you are looking straight ahead. You must lower (depress) your eyes to see the boat in the water. There are two possible ways to use our angle of depression to obtain an angle INSIDE the triangle. 1. Find the angle adjacent (next door) to our angle which is inside the triangle. This adjacent angle will always be the complement of our angle. Our angle and the angle next door will add to 90º. In the diagram on the left, the adjacent angle is 55º. 2. Utilize the fact that the angle of depression = the angle of elevation and simply place 35º in angle A. (the easiest method) Just be sure to place it in the proper position. SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction. Example 1: Practice: 1.) From a point on the ground 25 feet from the foot of a tree, the angle of elevation of the top of the tree is 32º. Find to the nearest foot, the height of the tree. 2.) SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction. 3.) Example 2: Refer to the triangle below: a) What is the relationship between mA and mB? ____________________ A 13 12 b) What is the cos A? _____ What is the sine B? _____ C 5 B c) What is the sin A? _____ What is the cos B? _____ What do you notice about the cosine and sine of complements? _____________________________________________________________ The sine of an acute angle is equal to the cosine of its complement. The sine and cosine functions are called cofunctions. SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction. Any trigonometric function of an acute angle is equal to the cofunction of its complement. cos x = sin (90o - x) sin x = cos (90o - x) o o Forsec example, = csc (90 - find) the value csc of = sin sec 30(90o. - ) Now find the value of cos 60o. SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction. Exit Ticket: Summary: SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction. Homework: 1. Find the value of that makes each statement true. a. ( ) b. ( ) c. ( ) d. ( ) 2. a. Make a prediction about how the sum will relate to the sum . b. Use the sine and cosine values of special angles to find the sum: . c. Find the sum . d. Was your prediction a valid prediction? Explain why or why not. 3. Langdon thinks that the sum is equal to . Do you agree with Langdon? Explain what this means about the sum of the sines of angles. 4.) 5.) 6.) SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction. Summary: SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine in terms of its CoFunction. SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known. Day 3 Using Trigonometry to Determine Area Warm Up: Determine the Area for each triangle below, if possible. If not possible, state what additional information would be needed. Triangle #1 Triangle #2 Triangle #3 *If there is missing information, is there a way to find the missing information? SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known. Example 1: What if the third side length of the triangle were provided? Is it possible to determine the area of the triangle now? Find the area of . (Hint: Draw Altitude HJ) SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known. Discussion: Let’s look at which is set up similarly to the triangle in Example 1: What formula would represent the area of this triangle? A= ½ ah Let’s rewrite the equation. Describe what you see: A= ½ ah _________________________________ A= ½ ab _________________________________ Look at below with an arc mark at vertex labeled . What do you notice about the structure of ? Can we think of this newly written area formula in a different way using trigonometry? Area of a Triangle: (Circle the part that represents the height) Using the new formula, what information would you need to calculate the area of a triangle? SWBAT: Derive the formula for calculating the Area of a Triangle when the height is not known. Example 2: A farmer is planning how to divide his land for planting next year’s crops. A triangular plot of land is left with two known side lengths measuring and . How could the farmer calculate the area of the plot, if the included angle between the known side lengths is 30⁰? You Try It: A real estate developer and her surveyor are searching for their next piece of land to build on. They each examine a plot of land in the shape of . The real estate developer measures the length of and and finds them to both be approximately feet, and the included angle has a measure of approximately .