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Transcript of Today – Monday, March 4, 2013 Warm Up: Which method to use to solve missing angles or sides of a...
Today – Monday, March 4, 2013
Warm Up: Which method to use to solve missing angles or sides of a right triangle
Review: Using inverse trig to find missing angles
Learning Target : Review solving for missing sides of right triangles using special triangles and trigonometry
QUIZ: Trigonometry and special right triangles
Independent Practice
30 °2𝑦
WARM UP: Decide which method would you use to find the missing sides of the right triangle (Pythagorean Theorem, Special Right Triangles, or Trig), then find the missing sides.
𝑥
621° 𝑦𝑥 9
7𝑥
METHOD TO FIND TRIG ANGLES:
𝑜𝑝𝑝𝑜
𝑠𝑖𝑡𝑒 h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡𝜃
“”
θ=𝑡𝑎𝑛− 1(𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 )
𝑺𝑶𝑯𝑪𝑨𝑯𝑻𝑶𝑨
1. Find the opposite, adjacent and hypotenuse of your right triangle.
θ=𝑐𝑜𝑠− 1( 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 )
θ=𝑠𝑖𝑛− 1( 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 )
2. Use the trig function that contains the sides of your triangle.3. Use the inverse trig function to calculate the angle.
912
PRACTICE: Find angle x using inverse trigonometric functions. Round your answer to the nearest hundredth.
𝑥
Use cosine ratio
Substitute known values
Use inverse cosine
Solve for x using calculator
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
8
14
PRACTICE: Find angle x using inverse trigonometric functions. Round your answer to the nearest hundredth.
𝑥
Use tangent ratio
Substitute known values Use inverse tangent
Solve for x using calculator
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝑠𝑖𝑑𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
13
57
PRACTICE: Find angle x using inverse trigonometric functions. Round your answer to the nearest hundredth.
𝑥
Use sine ratio
Substitute known values
Use inverse sine
Solve for x using calculator
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝑠𝑖𝑑𝑒
h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Solve for x using 8
6
PRACTICE: Find angle x using inverse trigonometric functions. Round your answer to the nearest hundredth.
𝑥 h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒10
Solve for y using
𝑦
There are multiple ways to solve this problem, you may
choose any trig function because all sides are given!
Inverse Trig Practice
15 minutes
We will check answers as a class.
Basic Right Triangle Problems:FINDING MISSING SIDES USE:• Pythagorean Theorem• Special Right Triangles:
30-60-9045-45-90
• Trig Ratios: sinx, cosx, tanx
FINDING MISSING ANGLES USE:• Inverse Trig:
:
√𝟐𝒙𝒙 45 °
45 °𝒙
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆=𝒍𝒆𝒈 ·√𝟐In a triangle:
𝟐 𝒙𝒙
60 °
30 °
√𝟑 𝒙
In a triangle:
METHOD TO FIND MISSING SIDES USING SPECIAL TRIANGLES:
h𝑠𝑜𝑟𝑡
𝑙𝑒𝑔
h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑙𝑜𝑛𝑔𝑙𝑒𝑔30 °
1. Determine which special triangle right triangle is shown-one angle will always be given. 2. Determine whether sides of the triangle that is given, is a leg (short or long or equal) or a hypotenuse. 3. Use given information from special triangle to plug into given equations, then solve for missing side.
𝒉𝒚𝒑
𝒆𝒒𝒖𝒂
𝒍𝒍𝒆𝒈
45 °
𝒆𝒒𝒖𝒂𝒍 𝒍𝒆𝒈
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆=𝒍𝒆𝒈 ·√𝟐
𝑥 9√2
PRACTICE: Find the value of x. Write your answer in the simplest radical form.
𝑥
Substitute known values Divide by
Simplify
20
𝑥
PRACTICE: Find the value of x and y. Write your answer in simplest radical form.
30 °
60 °
𝑦
Substitute known values Divide Simplify
Substitute known values
Simplify
√3 𝑥
PRACTICE: Find the value of x and y. Write your answer in simplest radical form.
30 °
60 °
𝑦 Substitute known values
Multiply
Substitute known values
Multiply Simplify
¿2√3
¿3
3√2 𝑥
PRACTICE: Find the value of x. Write your answer in the simplest radical form.
3√2
Substitute known values Multiply square roots
Simplify
TRIGONOMETRIC RATIOS:
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡𝜃
“”𝑆𝐼𝑁𝐸 : 𝑠𝑖𝑛𝜃=𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
=𝑂𝐻
𝐶𝑂𝑆𝐼𝑁𝐸 :𝑐𝑜𝑠 𝜃=𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
=𝐴𝐻
𝑇𝐴𝑁𝐺𝐸𝑁𝑇 :𝑡𝑎𝑛𝜃=𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
=𝑂𝐴
𝑺𝑶𝑯𝑪𝑨𝑯𝑻𝑶𝑨
METHOD TO FIND MISSING SIDES USING TRIGONOMETRY:
𝑜𝑝𝑝𝑜
𝑠𝑖𝑡𝑒 h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡𝜃
“”
1. Find the opposite, adjacent and hypotenuse of your right triangle. 2. Use the trig function that contains the sides of your triangle to set up your equation.3. Solve for x, then use your calculator to evaluate for the missing side.
𝑥
52
PRACTICE: Find x using the tangent ratio. Round answer to the nearest tenth.
26 °
Use tangent ratio Substitute known values Multiply by x on both sides
Divide by , both sides Solve for x using a calculator
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
67 𝑥
PRACTICE: Find x using the tangent ratio. Round answer to the nearest tenth.
46 °
Use sine ratio
Substitute known values
Multiply by 67 on both sides
Solve for n using calculator
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝑠𝑖𝑑𝑒h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
40
𝑥
PRACTICE: Find x using the cosine ratio. Round answer to the nearest tenth.
65 °
Use cosine ratio Substitute known values Multiply by x on both sides
Divide by , both sides Solve for x using a calculator
h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
Solve for x using 𝑥
6
PRACTICE: Find x and y using the tangent ratio. Round answer to the nearest tenth.
52 ° h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝑠𝑖𝑑𝑒
𝑦𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
Solve for y using
HOMEWORK:
If finished, work on other assignments:
HW #1: Pg. 436: 3-29 oddHW #2: Pg. 444: 1-6, 8, 10, 12HW #3: Pg. 461: 3-18HW #4: 7.4 Special Triangles WS (Kuta)HW #5: Pg. 470: 18-20, 24-29, 31, 32
Pg. 477: 10-15, 18-27HW #6: Pg. 485: 3-8, 10-18