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Apply the Tangent Ratio Chapter 7.5

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Trigonometric Ratio A trigonometric ratio is a ratio of 2 sides of a right triangle. You can use these ratios to find sides lengths and angle measures.

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Sides of a right triangle Opposite – the side opposite the angle you are looking at.

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Sides of a right triangle Adjacent – the side next to the angle you are looking at.

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Sides of a right triangle Hypotenuse – the side opposite the right angle. It is also the longest side on a triangle.

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Which side does the 22 represent? The hypotenuse, adjacent, or opposite?

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Which side is which?

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Tangent The ratio that we’ll focus on today is the tangent. The tangent is the opposite side over the adjacent side.

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Find the tangent of ŸR and ŸS To find the measure of the angle R, find the tangent. On a scientific calculator use the inverse tangent button to calculate the angle measure.

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Find the tangent of ŸJ and ŸK

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Find the Tangent of ŸA and ŸB, then the angle measures. Tan A = 0.75 Tan B = 1.333 m óA = 36.87ô m óB = 53.12ô Tan A = 1.05 Tan B = 0.95 m óA = 46.4ô m óB = 43.5ô Tan A = 0.4166 Tan B = 2.4 m óA = 22.62ô m óB = 67.38ô Tan A = 3.43 Tan B = 0.29 m óA = 73.75ô m óB = 16.17ô Tan A = 1.61 Tan B = 0.622 m óA = 58.15ô m óB = 31.88ô

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Finding missing side lengths Some problems may require you to find a missing side length. In these problems you will be given a side length and a measure of an angle. You will then use the fact that the tangent of an angle is equal to the opposite side over the adjacent side to find the angle.

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Example Multiply both sides by the denominator!

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Example Multiply both sides by the denominator!

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Example Multiply both sides by the denominator!

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Example Multiply both sides by the denominator! Divide both sides by the tangent!

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Example Multiply both sides by the denominator! Divide both sides by the tangent!

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Example Multiply both sides by the denominator! Divide both sides by the tangent!

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Example Multiply both sides by the denominator! Divide both sides by the tangent!

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Find the length of x for each problem. 1.2. 3. 4. X = 8.66 X = 21.98 X = 42.84 X = 25

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Tangents and “Special Right Triangles” Recall that for a 45-45-90 triangle the side lengths are: – leg = x-or- leg = 1 – Hypotenuse = x-or- Hypotenuse = Recall that for a 30-60-90 triangle the side lengths are: – Shorter leg = x-or- Shorter leg = 1 – Longer leg = x-or- Longer leg – Hypotenuse = 2x -or- Hypotenuse =2

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What length must x be?

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What must x be?

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A little more abstract… If I tell you that a right triangle has a measure of 30 degrees, could you find the tangent of the angle?

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A little more abstract… If I tell you that a right triangle has a measure of 45 degrees, could you find the tangent of the angle?

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5.2 Trigonometric Ratios in Right Triangles. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle.

5.2 Trigonometric Ratios in Right Triangles. A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle.

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