Geometry Right Triangles and Trigonometry

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1 Geometry Right Triangles and Trigonometry Day Date lass Homework Th 2/16 F 2/17 N: Special Right Triangles & Pythagorean Theorem Right Triangle & Pythagorean Theorem Practice Mid-Winter reak WKS: Special Right Triangles & Pythagorean Theorem Finish M 2/27 N: Trigonometric Ratios WKS: Trigonometry T 2/28 : Trig Mini ooklet (not included) G: asic Trigonometry W 3/1 N: Inverse Trigonometric Ratios WKS: Daffynition Decoder Th 3/2 N: Solving Right Triangles WKS: Solving Right Triangles F 3/3 M 3/6 : Solving Right Triangles Matching (not included) Quiz: asic Trig N: Trig Ratios in pplication T 3/7 N: ngle of Elevation & Depression W 3/8 Th 3/9 F 3/10 Half day : ngle of Elevation & Depression (not included) Review (not included) Quiz: ngle of Elevation & Depression G: Solving Right Triangles WKS: Finding Perimeter WKS: ngle of Elevation & Depression WKS: Elevation & Depression Study for Quiz Finish Review M 3/13 Go over Review & Review ctivity Study for Test T 3/14 Test

3 Name: Notes Spec. Rt. Triangles & Pythag. Thm. Standard: Period: Th PYTHGOREN THEOREM For any right triangle, a 2 + b 2 = c 2, where a and b are the lengths of the legs and c is the length of the hypotenuse. If a, b, and c are whole numbers, then they are considered a. Examples Find the length of the unknown side c = 15, b = 5 Will the following lengths create a right triangle? If so, do they form a Pythagorean triple? 3. 12, 16, , 8, and , 60, and 76

45 O -45 0-90 O TRINGLE Recall: an isosceles

hypotenuse is 2 times as long as each leg.

triangle, the hypotenuse is twice as long as

4 45 O O TRINGLE Recall: an isosceles triangle has two sides congruent and the angles opposite those sides congruent. 45 O O triangle is an isosceles triangle. In a 45 O O triangle, the hypotenuse is 2 times as long as each leg. Examples Find the length of the unknown side(s) O O TRINGLE In a 30 O O triangle, the hypotenuse is twice as long as the shortest side and the longer leg is 3 times as long as the shortest side. Examples Find the length of the unknown side(s)

5 Name: WKS Spec. Rt. Triangles & Pythag. Thm. Standard: Period: Determine if the set of measures below form a right triangle. If so, are a Pythagorean Triple? 1. 30, 40, , 30, , 4, Th 4. 5, 12, , 40, 41 Find the value of each variable in the following problems

6 For problems 15-18, read the problem, draw a picture, and complete the problem as directed. 15. Find the perimeter of a square with diagonal 12 cm. 16. Find the diagonal of a square with perimeter 20 inches. 17. The perimeter of an equilateral triangle is 33 cm. Find the length of an altitude of the triangle. 18. n altitude of an equilateral triangle is 5 3 meters. Find the perimeter of the triangle.

7 Name: Right Triangle & Pythagorean Thm. Practice Standard: Period: F Determine if the following sides form a right triangle. If so, state if they form a Pythagorean Triple , 80, , 60, , 4, 20 Use special right triangles to find the missing sides

8 x 6 2 y

9 Name: Notes Trigonometric Ratios Standard: Period: M DEFINITION is a ratio of the lengths of two sides in a right triangle. The of an angle is the ratio of the leg opposite the angle and the hypotenuse The of an angle is the ratio of the leg adjacent the angle and the hypotenuse The of an angle is the ratio of the legs of a right triangle. sinθ = opposite side hypotenuse SOH H TO adjacent side cos(θ) = hypotenuse tan(θ) = opposite side adjacent side adjacent side opposite side Examples Label the opposite and adjacent sides of each triangle and the hypotenuse Find each of the trigonometric ratios for each of the triangles. 4. K 5. Y 6. R sin(k) = sin(y) = sin(r) = cos(k) = cos(y) = cos(r) = tan(k) = tan(y) = tan(r) =

10 Write an equation that can be used to find the value of each variable, then solve the equation to find the value of x

11 Name: WKS Trigonometry Standard: Period: SOH H TO 1. Label the triangle with opposite, adjacent, and hypotenuse sides. M 2. Fill in the trig ratios 13 5 sin() = sin() = cos() = cos() = 12 tan() = tan() = 3. Fill in the trig ratios sin() = sin() = 8 cos() = cos() = 6 tan() = tan() = 4. Find the value of the variable 5. Find the value of the variable

12 6. Find the value of the variables 7. Find the value of the variables Determine whether the following form a right triangle and if they are a Pythagorean Triple. 8. 5, 7, , 20, , 1.5, 1.7 Find the lengths of each unknown side. 11. m = 90 o, m = 30 o 12. m = 90 o, m = 45 o 4 3 8

13 Name: Graded ssignment asic Trigonometry Standard: Period: T Determine if the following side lengths form a right triangle and if they are a Pythagorean Triple , 0.28, , 24, 26 Right Triangle: Pythagorean Triple: Right Triangle: Pythagorean Triple: Find the missing side lengths for each special right triangle. Use exact answers (leave answers in radical form). 3. m = 90 o, m = 30 o 4. m = 90 o, m = 45 o = = = = 5. Fill in the trig ratios. 5 3 sin() = sin() = cos() = cos() = tan() = tan() =

14 6. Given that cos(θ) = 8, find sine and tangent of θ. Sketch the triangle, label θ, and find all side lengths first. 17 sin(θ) = tan(θ) = 7. Write an equation using one of the trigonometric ratios to find the length of each unknown side. Then find each side length. Round to the nearest hundredth. m = 90 o, m = 52 o 11 Equation to find : Equation to find : = =

15 Name: Notes Inverse Trigonometric Ratios Standard: Period: Inverse Trigonometric Ratios To solve for an angle measure, you need to use an inverse trig ratio. W Inverse Sine sin(θ) = O H sin 1 ( O H ) = θ or arcsin (O H ) = θ Inverse osine cos(θ) = H cos 1 ( H ) = θ or arccos ( H ) = θ Inverse Tangent tan(θ) = O tan 1 ( O ) = θ or arctan (O ) = θ Set up the appropriate trig ratio and find the measure of each angle. Round to the nearest tenth of a degree

17 Name: WKS Daffynition Decoder Standard: Period: W

19 Name: Notes Solving Right Triangles Standard: Period: DEFINITION To a right triangle means to find the measure of all of its and. You only need to know 2 side lengths or a side and an acute angle to solve a right triangle. Th You will almost always use the following 3 steps (not always in order): Examples Solve the right triangle o o

21 Name: WKS Solving Right Triangles Standard: Period: Solve each right triangle. Round side lengths to the nearest hundredth and angles to the nearest tenth. 1. m = = 32 m = = Th m = = 2. m = m = = = 8.47 m = = 6 3. m = = 52 m = = 43 o m = = 4. m = = 26 o m = = 8 m = =

23 Name: Graded ssignment Solving Right Triangles Standard: Period: F Solve each triangle. Round angles to the nearest tenth and side lengths to the nearest hundredth. 1. m = = 17 m = = 9 m = = 2. m = = 69 m = = 31 o m = =

25 Name: Notes Trigonometric Ratios in pplication Standard: Period: M Find the perimeter of each figure

26 4. You are wanting to attach a dog run cable to the corner of the house and the corner of the connected garage. The angle the cable makes with the house is 35 o and the garage is 11ft wide. bout how much cable do you need to purchase for the dog run?

27 Name: WKS Finding Perimeter Standard: Find the perimeter of each figure Period: M m 53 o 14 inches 36

29 Name: Notes ngle of Elevation & Depression Standard: Period: T DEFINITION The angle of is the angle your line of sight makes with a horizontal when the object is above you. The angle of is the angle your line of sight makes with a horizontal when the object is below you. The angle of elevation and the angle of depression are congruent. 1. You are skiing down a mountain that has an altitude of 1200 meters. The angle of depression is 21 o. bout how far do you ski down the mountain? 2. From the top of a fire tower, a forest ranger sees his partner on the ground at an angle of depression of 40 o. If the tower is 45 feet tall, how far is the partner from the base of the tower, to the nearest tenth of a foot?

31 Name: WKS ngle of Elevation & Depression Standard: Period: T 1. nursery plants a new tree and attaches a guide wire to help support the tree while its roots take hold. n eight foot wire is attached to the tree and to a stake in the ground. From the stake in the ground the angle of elevation of the connection with the tree is 42 o. Find to the nearest tenth of a foot the height of the connection point on the tree. 2. Find the shadow cast by a 10 foot lamp post when the angle of elevation of the sun is 58 o. Find the length to the nearest tenth of a foot. 3. ladder leans against a brick wall. The foot of the ladder is 6 feet from the wall. The ladder reaches a height of 15 feet on the wall. Find to the nearest degree, the angle the ladder makes with the wall. 4. n airplane is flying at a height of 2 miles above the ground. The distance along the ground from the airplane to the airport is 5 miles. What is the angle of depression from the airplane to the airport?

33 Name: WKS Elevation & Depression Standard: Period: Sketch a diagram, then solve each problem. 1. John wants to measure the height of a tree. He walks exactly 100 feet from the base of the tree and looks up. The angle of elevation is 33 o. How tall is the tree? W 2. building is 50 feet high. worker in the building notices that the angle of depression from his office, on the top floor, to his favorite sub shop is 41 o. How far apart are the worker s job and his favorite sub shop? 3. n airplane takes off 600 feet in front of a 60 foot building. t what angle of elevation must the plane take off in order to avoid crashing into the building? 4. sledding run is 280 yards long with a vertical drop of 18.5 yards. What is the angle of depression of the run?

34 5. n airplane passes directly over Seneca High School. student at herokee High School, which is 4.3 miles away, finds the angle of elevation of the plane to be 19 o. Find the altitude of the plane. 6. bird sits on top of a lamppost. The angle of depression from the bird to the feet of an observer standing away from the lamppost is 35 o. The distance from the bird to the observer is 25 meters. How tall is the lamppost? 7. illy s kite has a string 40 feet long and is flying 27 feet above his eye level. Find the angle of elevation of the kite. 8. utility worker is on the top of an office building that is 50 feet tall. His coworker is 30 feet from the base of the building. Find the angle of depression from the worker on top of the building to his coworker on the ground.

Radicals and Pythagorean Theorem Date: Per:

Math 2 Unit 7 Worksheet 1 Name: Radicals and Pythagorean Theorem Date: Per: [1-12] Simplify each radical expression. 1. 75 2. 24. 7 2 4. 10 12 5. 2 6 6. 2 15 20 7. 11 2 8. 9 2 9. 2 2 10. 5 2 11. 7 5 2