bellringer find the value of each variable. if your answer is not an integer, express it in simplest...
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![Page 1: Bellringer Find the value of each variable. If your answer is not an integer, express it in simplest radical form 1. y=11 2. y= 9 √2 3. 3.2√2 4. y=7 √3](https://reader036.vdocuments.site/reader036/viewer/2022082418/5697bf991a28abf838c91d5c/html5/thumbnails/1.jpg)
Bellringer Find the value of each variable. If your answer is not an integer, express it in simplest radical form
• 1. y=11
• 2. y=9√2
• 3. 3.2√2
• 4. y=7√3
• 5. y=8
• 6. x=18
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Sect 8-3 TrigonometryGeometry: Chapter 8 Right Triangles and Trigonometry
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ReviewReview
• From the Pythagorean Theorem, similar triangles, and right triangles we learned the lengths of corresponding sides in similar right triangles have constant ratios.
• By the right and special right triangles we lead into the study of right triangle trigonometry.
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Lesson’s PurposeObjective Essential question
• To use the sine, cosine, and tangent ratios to determine the lengths.
• To use the sine, cosine, and tangent ratios to find the angle measures of right triangles.
• What are trigonometric ratios?
• Trigonometric ratios express relationships between the legs and the hypotenuse of a right triangle.
• Sine and cosine tell you the ratio of each leg to the hypotenuse.
• Tangent tells you the ratio of these legs to each other.
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Understanding Definition Real World Examples
• The word ‘trigonometry’ means ‘triangle measurement’.
• • Trigonometry involves the
ratios of the sides of right triangles.
• The three ratios are called tangent, sine and cosine.
• Trigonometry is an important tool for evaluating measurements of height and distance. It plays an important role in surveying, navigation, engineering, astronomy and many other branches of physical science.
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The Right Triangle Need to Know:
Labeling:
• Before we study trigonometry, we need to know how the sides of a right triangle are named.
• The three sides are called hypotenuse, adjacent and opposite sides.
• In the following right triangle PQR,
• the side PQ, which is opposite to the right angle PRQ is called the hypotenuse. (The hypotenuse is the longest side of the right triangle.)
• the side RQ is called the adjacent side of angle θ .
• the side PR is called the opposite side of angle θ
Note: The adjacent and the opposite sides depend on the angle θ . For complementary angle of θ , the labels of the 2 sides are reversed.
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Example #1Problem Solution:
• Identify the hypotenuse, adjacent side and opposite side in the following triangle:
• a) for angle x• b) for angle y
• a) For angle x: AB is the hypotenuse, AC is the adjacent side , and BC is the opposite side.
• b) For angle y: AB is the hypotenuse, BC is the adjacent side , and AC is the opposite side.
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Trigonometric Ratio: TangentTangent Ratio
• The tangent of an angle is the ratio of the opposite side and adjacent side.
• Tangent is usually abbreviated as tan.
• Tangent θ can be written as tan θ .
hypotenuse
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Example #2Calculate the value of tan θ in the following triangle.
Solution
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Trigonometric Ratio: SineRatio Sine
• The sine of an angle is the ratio of the opposite side to the hypotenuse side.
• Sine is usually abbreviated as sin. Sine θ can be written as sin θ .
hypotenuse
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Example #3
Calculate the value of sin θ in the following triangle.
Solution
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Trigonometric Ratio : CosineCosine Ratio
• The cosine of an angle is the ratio of the adjacent side and hypotenuse side.
• Cosine is usually abbreviated as cos. Cosine θ can be written as cos θ .
hypotenuse
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Example #4
SolutionCalculate the value of cos θ in the following triangle.
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Helpful Hint
•You may use want to use some mnemonics to help you remember the trigonometric functions.
•One common mnemonic is to remember the Indian Chief SOH-CAH-TOA.
•SOH Sine = Opposite over Hypotenuse. •CAH Cosine = Adjacent over Hypotenuse. •TOA Tangent = Opposite over Adjacent.
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Inverse Operations
• Now we are going to learn how to use these trigonometric ratios to find missing angles.
• Remember,• Addition and subtraction • Multiplication and Division
are inverse operation.
• One operation reverses the result you get from the other.
• Thus, if you know the sine, cosine, and tangent ratio for an angle, you can use the inverse (sinˉ¹,cosˉ¹,tan ˉ¹) to find the measures of the angles.
• If tan θ = x then tan -1 x = θ
• If sin θ = x then sin -1 x = θ
• If cos θ = x then cos -1 x = θ
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Example#5• Find the values of θ for
the following (Give your answers in degrees and minutes):
• a) tan θ = 2.53 • b) sin θ = 0.456 • c) cos θ = 0.6647
• Solution: • a) Press
• tan -1 2.53 = 68˚ 25 ’ 59.69 ” ( The ” symbol denotes seconds. There are 60 seconds in 1 minute.)
• = 68˚ 26 ’ (to the nearest minute) • b) Press
•sin -1 0.456 = 27˚ 7 ’ 45.46 ”
• = 27˚ 8 ’ (to the nearest minute) • b) Press
•cos -1 0.6647 = 48˚ 20 ’ 26.47 ”
• = 48˚ 20 ’ (to the nearest minute)
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Example #6• Calculate the angle x in
the figure below. Give your answer correct to 4 decimal places.
• Solution:
• sin x =
• x = sin -1(2.3 ÷ 8.15) • = 16.3921˚
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Real World Connections
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Summary Essential Knowledge
• We can use trigonometric functions to find both missing side measurements and to find missing angle measures.
• To find missing angles you must use the inverse trigonometric ratios.
• Once you know the right ratio, you can use your calculator to find correct angle measures.
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Ticket Out and HomeworkTicket Out Homework:
• If you know the lengths of two legs of a right triangle, how can you find the measures of the acute angles?
• Write a tangent ratio, then use inverses
• Pg 534-5 #’s 8-20• Pg538-9 #’s 4-14