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D. Trigonometry Math 10: Foundations and Pre-Calculus FP10.4 Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles.

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Key Terms: Find the definition of each of the following terms: Angle of Inclination Tangent Ratio Sine Ratio Cosine Ratio Indirect Measurement Angle of Elevation Angle of Depression

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1. The Tangent Ratio FP10.4 Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles.

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1. The Tangent Ratio Remember the Tan ratio? What is the tan ratio and what do we use it for?

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The value of the tangent ratio is usually expressed as a decimal that compares the lengths of the sides

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Example

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You can use a scientific calculator to determine the measure of an acute angle when you know the value tan ratio The tan -1 or Inv tan on your calculator does this for you

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Example

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Example

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Example

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Practice Ex. 2.1 (p. 74) #1-20 #6-23

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2. Calculating Length with Tangent Ratio FP10.4 Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles.

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The tangent ratio is a powerful tool we can use to calculate the length of a leg of a right triangle We are measuring indirectly when we measure this way

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We can find the length of a leg of a triangle by setting up the tangent formula, as long as we have one of the acute angles and the legs

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Example

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Example

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Example 3

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Practice Ex. 2.2 (p. 81) #1-14 #1-4, 6-16

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3. Sine and Cosine Ratios FP10.4 Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles.

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In a right triangle, the ratios that relate each leg to the hypotenuse depend only on the measure of the acute angle not the size of the triangle These ratios are called the sine and cosine ratios

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The sine ratio is written sin θ The cosine ratio is written cos θ

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The sine, cosine and tangent ratios are called the primary trig ratio The values of the trig ratios are often expressed as decimals

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Example

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Example

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Example

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Practice Ex. 2.4 (p. 94) #1-15 #1-3, 5-17

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4. Using Sine and Cosine to find Length FP10.4 Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles.

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4. Using Sine and Cosine to find Length

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Construct Understanding p. 97

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We can use the sin and cos ratios to write an equation that we can solve to calculate the length of a leg in a right triangle When the measure of one acute angle and the hypotenuse are know

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Example

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The sin and cosine ratios can be used to calculate the measure of the hypotenuse When the measure of one acute angle and the length of one of the legs are known

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Example

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Example

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Practice Ex. 2.5 (p. 101) #1-12 #1-14

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5. Applying Trig FP10.4 Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles.

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5. Applying Trig

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Construct Understanding p. 105

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When we calculate the measures of all the angles and all the side lengths in a right triangle, we solve the triangles.

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We can use any of the three primary trig ratios to do this.

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Example

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Example

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Example

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Practice Ex. 2.6 (p. 110) #1-14 #1-2, 5-16

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6. Problems with More Triangles FP10.4 Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles.

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6. Problems with More Triangles We can use Trig to solve problems that can be modeled using right triangles When one more right triangle is involved, we have to decide which triangle to start with

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Example

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Example

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Sometimes the right triangles are not even in the same plane

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Example

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Practice Ex. 2.7 (p. 118) #1-14 #5-21

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EXAMPLE 1 Finding Trigonometric Ratios For PQR, write the sine, cosine, and tangent ratios for P. SOLUTION For P, the length of the opposite side is 5.

EXAMPLE 1 Finding Trigonometric Ratios For PQR, write the sine, cosine, and tangent ratios for P. SOLUTION For P, the length of the opposite side is 5.

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