Unit 3

Right Triangles trigonometryGSE GEOMETRYpEBBLEBROOK HIGH SCHOOL | mATH DEPARTMENT

Vocabulary BuilderChoose the concept from the list below that best represents the item in each box.

Draw a line from each word in Column A to its definition in Column B.10. Sine11. Cosine12. Tangent

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3 - 1 Trigonometric RatiosIf the ____________________________ of a number is a whole number, the original number is called a __________________________________, which is a number multiplied by itself.

A _______________________________________________ is an expression that contains a square root. The number under the radical sign is called the __________________________________.To simplify a radical expression, make sure that the radicand has no __________________________ factors other than 1.Example 1Simplify each expression.

Simplify each expression.

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The Pythagorean Theorem is probably the most famous mathematical relationship. The theorem states that in a ______________________________, the sum of the squares of the lengths of the legs _________________ the square of the length of the ___________________________________.The Pythagorean Theorem given you a way to find unknown ____________________________ when you know a triangle is a right triangle.Example 2Find the value of x. Give your answer in simplest radical form.

Find the value of x. Give your answer in simplest radical form.

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We will further study right triangles by looking at trigonometric values as defined by ratios of the sides of a right triangle. The side labeled ________________________ is always opposite the right angle of the right triangle. The other two sides of the right triangle are determined by the angle that is being discussed. The __________________________ side will always make up part of the angle that is being discussed and cannot be the hypotenuse. The side of the right triangle that DOES NOT form part of the discussed angle is called the _______________________ side.Example 3

Identify the opposite, adjacent, and hypotenuse of the following triangles.

Identify the opposite, adjacent, and hypotenuse of the following triangles.

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A _______________________________________ is a ratio of two sides of a right triangle. Thinking _________________can help you remember these ratios.

Using the ratios below, you can find the ________________ of any side of a ________________ triangle if you know one ________________ angle and any other side.

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Example 3Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth.

Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth.

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Simplify the square root.

Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth.

Simplify each square root in simplest radical form.

Identify the sides of the right triangle below.

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Find the value of each trigonometric ratio.

Find the missing side of each triangle. Leave your answers in simplest radical form.

3 - 2 Finding Missing Sides and Angles of a Right TriangleSine, Cosine, and Tangent are trigonometric __________________. The __________________ of each function is a(n) ______________________ measure. For each trigonometric function, every acute angle measure produces a different __________________________, or value of the function.

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Example 1

Find each length. Round to the nearest hundredth.

Find each missing length. Round to the nearest hundredth.

Using an inverse trigonometric function, such as __________, ___________, __________ allows you to determine an unknown angle measure given sides of right triangle.Example 2

Find the missing measure. Round to the nearest tenth.10 | P a g e

1.

Find the missing measure. Round to the nearest tenth.

Find each missing measure. Round to the nearest tenth.

Find each missing measure. Round to the nearest tenth.YYYYYY

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2. 3.

4. 5. 6.

1. 2.

1.2. 3.

3.

x˚

3 – 3 Sine and Cosine of Complementary AnglesThe sum of the measures of the interior angles of a triangle is ____________. Every right triangle has one right angle, so the sum of the measures of the two acute angles in any right triangle must 12 | P a g e

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be equal to ___________. Angles that add up to 90˚ ________________________________ angles.In a right triangle, the __________________________________ for one acute angle is the adjacent leg for the other acute angle. So, the sine of one acute angle is equal to the _______________________ of its complement, and vice versa.

Example 1

Write each trigonometric function in terms of its complement.1. sin 64˚ 2. cos84 ˚ 3. cos38 ˚4. sin 24 ˚ 5. cos72 ˚ 6. sin 45 ˚

Write each trigonometric function in terms of its complement.1. sin 22˚ 2. cos65 ˚ 3. cos 44 ˚4. sin 32 ˚ 5. cos25 ˚ 6. sin 15 ˚Example 2

Find the missing values.13 | P a g e

sin A=cos (90−A)

cosB=sin (90−B)

What do you notice about the relationship between sine and cosine?__________________________________________________________________________________________Find the missing values.

What do you notice about the relationship between sine and cosine?14 | P a g e

a. sin A=¿¿ b. sin B=¿¿

b. cos A=¿¿ d. cosB=¿¿

a. sin A=¿¿ b. sin B=¿¿

b. cos A=¿¿ d. cosB=¿¿

a. sin K=¿¿ b. sin J=¿¿

b. cosK=¿¿ d. cos J=¿¿

a. sin S=¿¿ b. sinT=¿¿

b. cos S=¿¿ d. cosT=¿¿

mP = 30˚

mS = 45˚

J

K

L

__________________________________________________________________________________________Example 3

Write each trigonometric expression.

Write each trigonometric expression.

Example 4

Draw ABC where ACB = 90˚. AC = 5 and CB = 12.a. What is the length of AB?b. What is cos A?c. What is sin B?Draw HAT where H = 90˚ and tan A=12

35.a. What is the length of AT?b. What is sin A?c. What is cos T?15 | P a g e

Draw XYZ where Y = 90˚. XY = 8 and YZ= 6.a. What is the length of XZ?b. What is cos X?c. What is sin Z?Draw MIX where I = 90˚ and cos X=3

5.a. What is the length of IM?b. What is sin X?c. What is cos M?

Write each trigonometric function in terms of its complement.1. sin 13˚ 2. cos66 ˚ 3. sin 75 ˚Find the missing values.

5. Draw BAD where A= 90˚ and cosD= 9

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a. sinW=¿¿ b. sinV=¿¿

b. cosW=¿¿ d. cosV=¿¿

4.17

15

W

VY

a. What is the length of IM?b. What is sin D? c. What is cos B?

Write each trigonometric function in terms of its complement.1. cos30˚ 2. sin 80 ˚ 3. cos 45 ˚4. sin 73˚ 5. cos26 ˚ 6. sin 25 ˚Find the missing values.

9. Draw PAT where A= 90˚ and cosP=45 .

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a. sin K=¿¿ b. sin L=¿¿

b. cosK=¿¿ d. cos L=¿¿

a. sin K=¿¿ b. sin L=¿¿

b. cosK=¿¿ d. cos L=¿¿

7.

8.

a. What is the length of AT?b. What is sin P? c. What is cos T?

3 – 4 Solve Right TrianglesNow that we know how to write the 3 trigonometric functions of a right triangle, we can use these ratios to _______________________ a right triangle. Solving a right triangle means that we use given _________________________ and ____________________________ measures to calculate missing sides and angle measures.

Example 1

Find ALL the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree.

Find ALL the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 18 | P a g e

a. RP=¿¿ b. ∠P=¿ ¿

c. ∠R=¿¿

a. BC=¿¿ b. AB=¿¿

c. ∠C=¿¿

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A(n) ______________________________________ is the angle formed by a horizontal line and the line of sight to an object ABOVE the horizontal line. A(n) ______________________________________is the angle formed by a horizontal line and the line of sight to an object BELOW that horizontal line.

Example 2

Describe each angle as it relates to the situation in the diagram.

Describe each angle as it relates to the situation in the diagram.19 | P a g e

Example 3Find the value of x. Round to the nearest tenth.

Find the value of x. Round to the nearest tenth.

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1. 2.

3.

Describe each angle as it relates to the situation in the diagram.

Find the value of x. Round to the nearest tenth.

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3.

5.

Find the value of x. Round to the nearest tenth.

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7. 8.

9.

1.

Describe each angle as it relates to the situation in the diagram.

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3. 4.

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