7.1 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right...

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Transcript of 7.1 Right Triangle Trigonometry. A triangle in which one angle is a right angle is called a right...

  • Slide 1
  • 7.1 Right Triangle Trigonometry
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  • A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called the hypotenuse, and the remaining two sides are called the legs of the triangle. c b a
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  • Find the value of each of the six trigonometric functions of the angle Adjacent 12 13 c = Hypotenuse = 13 b = Opposite = 12
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  • a b c To solve a right triangle means to find the missing lengths of its sides and the measurements of its angles.
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  • b 4 c
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  • 25h h = 23.49
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  • 7.2 The Law of Sines
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  • If none of the angles of a triangle is a right angle, the triangle is called oblique. All angles are acute Two acute angles, one obtuse angle
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  • To solve an oblique triangle means to find the lengths of its sides and the measurements of its angles.
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  • CASE 1: ASA or SAA S A A ASA S AA SAA
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  • S S A CASE 2: SSA
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  • S S A CASE 3: SAS
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  • S S S CASE 4: SSS
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  • The Law of Sines is used to solve triangles in which Case 1 or 2 holds. That is, the Law of Sines is used to solve SAA, ASA or SSA triangles.
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  • Theorem Law of Sines
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  • 7.3 Law of Cosines
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  • We use the Law of Sines to solve CASE 1 (SAA or ASA) and CASE 2 (SSA) of an oblique triangle. The Law of Cosines is used to solve CASES 3 and 4. CASE 3: Two sides and the included angle are known (SAS). CASE 4: Three sides are known (SSS).
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  • Theorem Law of Cosines Remember to give alternate form of law of cosines!
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  • 7.4 Area of a Triangle
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  • Theorem The area A of a triangle is where b is the base and h is the altitude drawn to that base.
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  • Theorem The area A of a triangle equals one-half the product of two of its sides times the sine of its included angle.
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  • Theorem Herons Formula The area A of a triangle with sides a, b, and c is
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  • Find the area of a triangle whose sides are 5, 8, and 11.
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  • Additional Examples Page 561: 25, 27, and 29
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  • 7.5 Simple Harmonic Motion; Damped Motion; Combining Waves
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  • Simple harmonic motion is a special kind of vibrational motion in which the acceleration a of the object is directly proportional to the negative of its displacement d from its rest position. That is, a = -kd, k > 0.
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  • Theorem Simple Harmonic Motion An object that moves on a coordinate axis so that its distance d from the origin at time t is given by either
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  • The frequency f of an object in simple harmonic motion is the number of oscillations per unit of time. Thus,
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  • Suppose an object is attached to a pendulum and is pulled a distance 7 meters from its rest position and then released. If the time for one oscillation is 4 seconds, write an equation that relates the distance d of the object from its rest position after time t (in seconds). Assume no friction.
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  • Suppose that the distance d (in centimeters) an object travels in time t (in seconds) satisfies the equation (a) Describe the motion of the object. Simple harmonic (b) What is the maximum displacement from its resting position? A = |-15| = 15 centimeters.
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  • Suppose that the distance d (in centimeters) an object travels in time t (in seconds) satisfies the equation (c) What is the time required for one oscillation? (d) What is the frequency? Period : frequencyoscillations per second.
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  • Theorem Damped Motion The displacement d of an oscillating object from its at rest position at time t is given by where b is a damping factor (damping coefficient) and m is the mass of the oscillating object.
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  • Suppose a simple pendulum with a bob of mass 8 grams and a damping factor of 0.7 grams/second is pulled 15 centimeters to the right of its rest position and released. The period of the pendulum without the damping effect is 4 seconds. (a) Find an equation that describes the position of the pendulum bob.
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  • (b) Using a graphing utility, graph the function. (c) Determine the maximum displacement of the bob after the first oscillation.
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  • Assignment Page 561: 10, 18, 26, 30, and 40 Page 571: 16, 18, and 28