terminal arm length and special case triangles day 2
TRANSCRIPT
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Terminal Arm Length and Special Case Triangles
DAY 2
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Using Coordinates to Determine Length of the Terminal Arm
• There are two methods which can be used: – Pythagorean Theorem – Distance Formula
• Tip: “Always Sketch First!”
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Using the Theorem of Pythagoras
• Given the point (3, 4), draw the terminal arm. 1. Complete the right triangle by joining the
terminal point to the x-axis.
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Solution
2. Determine the sides of the triangle. Use the Theorem of Pythagoras.
• c2 = a2 + b2 • c2 = 32 + 42 • c2 = 25 • c = 5
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Solution continued
3. Since we are using angles rotated from the origin, we label the sides as being x, y and r for the radius of the circle that the terminal arm would make.
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Example: Draw the following angle in standard position given any point (x, y) and determine the value of r.
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Using the Distance Formula
The distance formula: d = √[(x2 – x1)2 + (y2 – y1)2]
• Example: Given point P (-2, -6), determine the length of the terminal arm.
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Review of SOH CAH TOA
• Example: Solve for x. • Example: Solve for x.
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Example: Determine the ratios for the following:
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Special Case Triangles – Exact Trigonometric Ratios
• We can use squares or equilateral triangles to calculate exact trigonometric ratios for 30°, 45° and 60°.
• Solution
• Draw a square with a diagonal. • A square with a diagonal will have angles of 45°. • All sides are equal. • Let the sides equal 1
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45°
• By the Pythagorean Theorem, r = 2
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30° and 60°
• All angles are equal in an equilateral triangle (60°)
• After drawing the perpendicular line, we know the small angle is 30°
• Let each side equal 2 • By the Theorem of
Pythagoras, y = 3
Draw an equilateral triangle with a perpendicular line from the top straight down
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Finding Exact Values
• Sketch the special case triangles and label • Sketch the given angle • Find the reference angle
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Example: cos 45°
1cos45
2
1 2
2 2
2
2
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Example: sin 60°
3sin 60
2
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Example: Tan 30°
• Example: Tan 30° • Example: Cos 30°
1 3tan30
33
3cos30
2
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Solving Equations using Exact Values, Quadrant I ONLY
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ASSIGNMENT:
• Text pg 83 #8; 84 #10, 11, 12, 13