Unit: Trigonometry and Unit Circle

Size: px

Start display at page:

Download "Unit: Trigonometry and Unit Circle"

Candace West
7 years ago
Views:

1 Unit: Trigonometry and Unit Circle NAME Per 5/ SR Finals nd, 5 th, 7 th Unit Circle HW: Part 5/ SR Finals 3 rd, 4 th, 6 th Unit Circle HW: Part st, nd, 5 th, 7 th 3 rd, 4 th, 6 th 5/3 SR Finals st No Advisory Quiz: Unit Circle Trig sin, cos, tan HW: Part st 7 th 5/6 HOLIDAY 5/7 Quiz: Unit Circle Applications of Trig HW: Part 3 5/8-9 Quiz: Unit Circle Trig sec, csc, cot HW: Part 4 & Review 5/30 TEST: Trig Test 6/ Final Exam Review 6/3 Final Exam Review Will sign exemptions! 6/4 Finals 6/5 Finals 6/6 Finals st, nd, 5 th 3 rd, 6 th 4 th, 7 th Objectives: To fill in the unit circle correctly. To solve for missing parts of a right triangle using trigonometry. To solve real world problems using trigonometry. To know the 3 reciprocal ratios of trig. Pre-AP: To use the unit circle to solve problems with exact answers. Essential Questions: How do you know which trigonometric ratio to use? What patterns can you use to build the unit circle?? How do the 6 trig ratios relate to each other?

2 Part Unit Circle The Special Right Triangles If we make the hypotenuse : 30 3 Next we ll take one of those triangles and put it on a coordinate plane: A What would the x- and y- coordinates of point A (formed by a 30 angle) be? (Hint: Use the lengths of the sides of the triangle.) (0,0) 30 Do the same for a 45 angle and a 60 angle: B C If you look at the three points we have traced out so far, they all lie on a circle centered at the origin. (0,0) 45 (0,0) 60 What is the radius of this circle?

3 DEGREES: RADIANS: Another way to measure angles. Coordinates: Memory Tricks: The test is the blank Unit Circle: Be able to fill it in by memory!!!!

4 Putting it all together: PRE-AP Use the work you did with the triangles on the front page to fill in the following table: Angle Coordinate Value of Value of Value of measure on unit circle sine cosine tangent What relationship do you notice between the coordinate on the unit circle and the values of sine and cosine? Tangent? 60 Working backwards on the unit circle: What is the location on the Unit Circle? 0

5 Part Trig Ratio st What MODE should your calculator be in? Use your calculator to estimate the following values. Round to the nearest hundredth. Ex. sin 35 = Ex. cos 8 = Ex 3. tan 87 = Use your calculator to estimate the following values. Round to the nearest degree. 8 Ex 7. sin α = α = Ex 8. tan β =.875 β = Ex 9. 7 cos θ = θ = There are 3 of trigonometric relationships that we study. Sine is the ratio of the side to the. Cosine is the ratio of the side to the. Tangent is the ratio of the side to the side. The NEVER changes, but and are dependent on the used. The angle is NEVER used. Memory Trick: The trigonometric ratios are written in an equation form. (**Hint: Write these ratios at the top of EVERY page you are working on.) Sineθ = Cosineθ = Τ angent θ = USE THE TRIANGLE AT THE RIGHT to determine the following trigonometric ratios. Ex 4. sin α = Ex 5. sin β = Ex 6. cos α = Ex 7. cos β = Ex 8. tan α = Ex 9. tan β = *PRE AP Solve using the unit circle. No decimal answers allowed! x 0 cos30 = sin 35 = 8 n 5 cos 0 = m

6 Use Trigonometric Ratios to Solve for Missing Sides and Angles ) Determine which Trig Ratio will fit your information. ) Set up the Trig Ratio 3) Round to the nearest degree if it is an angle and round to the nearest hundredth for sides. Ex. Ex. w k w = k = Ex 3. Ex 4. 4 x 6 x 57 8 x = x = Ex 5. Ex θ 7 β θ = β =

7 Part 3 Applications Angle of depression - Angle if elevation - Ex. Angie looks up at 5 degrees to see an airplane flying toward her. If the plane is flying at an altitude of 3.5 miles, how far is it from being directly above Angie? Ex. A six foot vertical pole casts a shadow of feet. What is the angle, β, of elevation with the ground? Ex 3. A 30 foot tree broke from its base and fell against a house. If the tree touches the house at feet, what angle is the tree forming with the house? Ex 3. Lauren is at the top of a 5 m lookout tower. From an angle of depression of 5, she sees Evan coming toward her. How far is Evan from the base of the tower? Ex 4. Christian is shooting a toy rocket off at an angle of 50 with the ground. He is 45 ft from his house and his house is 36 ft high. Will the rocket clear the house? Ex 5. At a point 75 ft from the base of a cliff, the angle of elevation to the first ridge is 30 and to the top of the cliff the angle of elevation is 60. How much higher is the top of the cliff as compared to the first ridge?

8 Part 4 Reciprocal Trig Functions You can also take the reciprocal of each trigonometric function. The Reciprocal Trigonometric Ratios are as follows: Reciprocal of Sine Function: Cosecant (csc) is the ratio of the side to the. Cosecant is also: cscθ = cscθ = sinθ H O Reciprocal of Cosine Function: Secant (sec) is the ratio of the side to the. Secant is also: Reciprocal of Tangent Function: secθ = secθ = cosθ H A Cotangent (cot) is the ratio of the side to the. Cotangent is also: A cotθ = cotθ = tanθ O Use the triangle at the right to determine the following ratios. Be sure to simplify your answers! Ex. csc θ =. csc 40 = θ Ex 3. sec θ = 4. sec 40 = 4 5 Ex 5. cot θ = 6. cot 40 = PRE-AP Use the Unit Circle to solve for x. No decimal answers!!! 3π 5π 8. csc = x 9. cot = x 0. tan( 0) = x º Memory Trick:

Find the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places.

Find the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places. SECTION.1 Simplify. 1. 7π π. 5π 6 + π Find the measure of the angle in degrees between the hour hand and the minute hand of a clock at the time shown. Measure the angle in the clockwise direction.. 1:0.