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How do we analyze the relationships between the angles and sides of right triangles?
Agenda: Warmup Notes/practice – sin/cos/tan Core Assessment 1 Monday Right Triangles TEST Tuesday
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Today’s objective is to set up trigonometric ratios.
So far this unit we have found side lengths in right triangles by either using the _________________ or by using the ______________________ Pythagorean Theorem special triangle relationships. Now we can find out all information about a right triangle (all side lengths and all angle measures) if we know: a) one side length and one angle or b) two side lengths. We use trigonometric ratios to accomplish this goal. Today’s objective is to set up trigonometric ratios.
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Hypotenuse Adjacent Opposite
Trigonometric functions are always set up based on two of the three sides of a right triangle. The sides are called Opposite, Adjacent, and Hypotenuse. x° Adjacent Hypotenuse Opposite
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Hypotenuse Adjacent Opposite
To set up the sine ratio, you compare the opposite side to the hypotenuse. x° Adjacent Hypotenuse Opposite
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Hypotenuse Adjacent Opposite
To set up the cosine ratio, you compare the adjacent side to the hypotenuse. x° Adjacent Hypotenuse Opposite
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Hypotenuse Adjacent Opposite
To set up the tangent ratio, you compare the opposite side to the adjacent side. x° Adjacent Hypotenuse Opposite
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There is an acronym to remember how to set up the trigonometric ratios:
SOHCAHTOA S C O T A O H H A
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Example 1 – Set up the 6 trig ratios for x and y.
5 4 y° 3 SOHCAHTOA
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Ex 2 – Set up the 6 trig ratios for 20 and ___.
70° y 70° 4 20° SOHCAHTOA x
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First use Pythagorean Theorem to find the missing side length.
Ex3 – Set up the 6 trig ratios for x and y. SOHCAHTOA First use Pythagorean Theorem to find the missing side length. 52 + z2 = 132 z2 = 144 z = 12 25 + z2 = 169 x° 13 5 y° 12
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We will look at two types of problems:
1) If you are given a right triangle with one angle measurement and one side length, then you can find the other two sides and last angle measurement. 2) If you are given a right triangle with two side lengths, then you can find the other side length and the two angle measurement.
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SOHCAHTOA Type 1 – Example 1 x 10 y sin(55) = cos(55) = tan(55) =
Step 2 – Set up the 6 possible trig ratios. sin(55) = cos(55) = tan(55) = sin(35) = cos(35) = tan(35) = Step 1 – Find the last angle measure. 35° x 10 55° y Step 3 – Choose two trig functions to use and solve for all variables. SOHCAHTOA Step 4 – Check your answers using the Pythagorean Theorem.
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SOHCAHTOA Type 1 – Example 2 x y 10 sin(53) = cos(53) = tan(53) =
Step 2 – Set up the 6 possible trig ratios. sin(53) = cos(53) = tan(53) = sin(37) = cos(37) = tan(37) = Step 1 – Find the last angle measure. 10 53° x 37° y Step 3 – Choose two trig functions to use and solve for all variables. SOHCAHTOA Step 4 – Check your answers using the Pythagorean Theorem.
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SOHCAHTOA Type 1 – Example 3 y 9 x sin(59) = cos(59) = tan(59) =
Step 2 – Set up the 6 possible trig ratios. sin(59) = cos(59) = tan(59) = sin(31) = cos(31) = tan(31) = Step 1 – Find the last angle measure. x 59° y 31° 9 Step 3 – Choose two trig functions to use and solve for all variables. SOHCAHTOA Step 4 – Check your answers using the Pythagorean Theorem.
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Type 1 – Example 3 - Revisited
30° 60° 90° n 2n x 60° y 30° 9 y: x:
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