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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 Trigonometry Geometry: Chapter 8 Right Triangles and Trigonometry

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Review 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 Purpose ObjectiveEssential 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 DefinitionReal 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 #1 ProblemSolution: 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: Tangent Tangent 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 #2 Calculate the value of tan θ in the following triangle. Solution

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Trigonometric Ratio: Sine RatioSine 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 : Cosine CosineRatio 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 Solution Calculate 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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Questions to ask? 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. How could you determine the value of 35˚ without using a calculator? Draw a right triangle with one acute angle equally to 35˚ then measure the lengths of the opposite side and the hypotenuse. Finally, find the ratio of the opposite and hypotenuse.

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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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Why it Works! Each angle measure in a right triangle corresponds to a given ratio. If the given ratio is opposite over hypotenuse, for example, the inverse of sine gives the correct angle measure. How do we know why this will work? What do we need to know why it works?

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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 Homework Ticket OutHomework: 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

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