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Trigonometric Ratios Please view this tutorial and answer the follow-up questions on loose leaf to turn in to your teacher.

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Identifying Parts of a Right Triangle Hypotenuse – always across from the 90° angle Hypotenuse – always across from the 90° angle Side Opposite – always across from the angle being referenced Side Opposite – always across from the angle being referenced Side Adjacent- always touching the angle being referenced Side Adjacent- always touching the angle being referenced *Note that all angles are marked with capitol letters and sides are marked with lower case letters *Note that all angles are marked with capitol letters and sides are marked with lower case letters A BC Angle C measures 90°

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Identifying Parts of a Right Triangle What side is opposite of angle A? What side is opposite of angle A? Side BC Side BC What side is opposite of angle B? What side is opposite of angle B? Side AC Side AC What side is adjacent to angle A? What side is adjacent to angle A? Side AC Side AC What side is adjacent to angle B? What side is adjacent to angle B? Side BC Side BC What side is the hypotenuse? What side is the hypotenuse? Side AB Side AB A BC

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Trigonometric Ratios (only apply to right triangles) Sine (abbreviated sin) Sine (abbreviated sin) Sin x° = Sin x° = Example: Example: A CB Sin A =

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Trigonometric Ratios (only apply to right triangles) Cosine (abbreviated cos) Cosine (abbreviated cos) Cos x° = Cos x° = Example: Example: A CB Cos A =

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Trigonometric Ratios (only apply to right triangles) Tangent (abbreviated tan) Tangent (abbreviated tan) Tan x° = Tan x° = Example: Example: A CB Tan A =

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Helpful Hint to Remember the Trig Ratios SOH (sine = opposite / hypotenuse) SOH (sine = opposite / hypotenuse) CAH (cosine = adjacent / hypotenuse) CAH (cosine = adjacent / hypotenuse) TOA (tangent = opposite / adjacent) TOA (tangent = opposite / adjacent) Remember SOH CAH TOA Remember SOH CAH TOA

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Time to Practice Identify the following trig ratio values Identify the following trig ratio values C B A Sin A =Sin B= Cos A = Cos B= Tan A = Tan B= 3 4 5

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Time to Practice Identify the following trig ratio values Identify the following trig ratio values C B A Sin A =Sin B= Cos A = Cos B= Tan A = Tan B= 3 4 5

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More Practice Identify the following trig ratio values Identify the following trig ratio values C B A Sin A =Sin B= Cos A = Cos B= Tan A = Tan B=

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More Practice More Practice Identify the following trig ratio values Identify the following trig ratio values C B A Sin A =Sin B= Cos A = Cos B= Tan A = Tan B=

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How to use the trig ratios to find missing sides Step 1: Make sure your calculator is in degree mode Step 1: Make sure your calculator is in degree mode Step 2: Label the right triangle with the words opposite, adjacent, and hypotenuse based on the given angle (Note: Do not use the right angle.) Step 2: Label the right triangle with the words opposite, adjacent, and hypotenuse based on the given angle (Note: Do not use the right angle.) Step 3: From the given information, determine which trig ratio should be used to find the side length Step 3: From the given information, determine which trig ratio should be used to find the side length Step 4: Substitute in the given information Step 4: Substitute in the given information

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How to use the trig ratios to find missing sides (continued) Step 5: Put a 1 under the trig ratio Step 5: Put a 1 under the trig ratio Step 6: Cross multiply Step 6: Cross multiply Step 7: When x=, put problem into your calculator (Note: you may have to divide first to get x by itself) Step 7: When x=, put problem into your calculator (Note: you may have to divide first to get x by itself) (NOTE: The angles of a triangle MUST add up to be 180°) (NOTE: The angles of a triangle MUST add up to be 180°)

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Example Given the following triangle, solve for x. Given the following triangle, solve for x. 8 cm x 60°

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Let’s Talk Through the Steps Step 1 : Check calculator for degree mode Step 1 : Check calculator for degree mode Press the Mode button and make sure Degree is highlighted as in the picture below Press the Mode button and make sure Degree is highlighted as in the picture below

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Step 2 Label the triangle according to the given angle Label the triangle according to the given angle 8 cm- HYPOTENUSE X - OPPOSITE 60°

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Step 3 Identify the trig ratio we should use to solve for x. Identify the trig ratio we should use to solve for x. 8 cm- HYPOTENUSE X - OPPOSITE 60° From the 60° angle, we know the hypotenuse and need to find the opposite. So we need to use SINE.

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Step 4 Substitute in the given information into the equation. Substitute in the given information into the equation. 8 cm- HYPOTENUSE X - OPPOSITE 60° Sin x°= Sin 60° =

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Step 5 Put a 1 under the trig ratio Put a 1 under the trig ratio 8 cm- HYPOTENUSE X - OPPOSITE 60° Sin x°= Sin 60° = 1

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Step 6 Cross multiply to solve for x Cross multiply to solve for x Sin 60° = 1 8 sin (60°) = x

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Step 7 Since x is already by itself, I can enter the information into the calculator. Since x is already by itself, I can enter the information into the calculator. Therefore, we can state that x=6.93.

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Let’s Look at Another Example Suppose that when we set-up the ratio equation, we have the following: Suppose that when we set-up the ratio equation, we have the following: Tan 20° = Tan 20° =

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What Happens When We Cross Multiply? Tan 20° = Tan 20° = 1 X tan 20° = 4 (How do we get x by itself?) tan 20° tan 20° (Now we have to divide by tan 20° in order to solve for x) tan 20° tan 20° (Now we have to divide by tan 20° in order to solve for x) X = 4 tan 20° tan 20° X = 10.99

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How to use the trig ratios to find missing angles Step 1: Make sure your calculator is in degree mode (See slide 15) Step 1: Make sure your calculator is in degree mode (See slide 15) Step 2: Label the right triangle with the words opposite, adjacent, and hypotenuse based on the given angle (Note: Do not use the right angle.) Step 2: Label the right triangle with the words opposite, adjacent, and hypotenuse based on the given angle (Note: Do not use the right angle.) Step 3: From the given information, determine which trig ratio should be used to find the side length Step 3: From the given information, determine which trig ratio should be used to find the side length Step 4: Substitute in the given information Step 4: Substitute in the given information

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How to use the trig ratios to find missing sides (continued) Step 5: Solve for x by taking the inverse (opposite operation) of the trig ratio. Step 5: Solve for x by taking the inverse (opposite operation) of the trig ratio. Step 6: When x=, put problem into your calculator. Step 6: When x=, put problem into your calculator.

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Calculator Steps for Finding Angles To solve for x, remember to take the inverse trig function. To solve for x, remember to take the inverse trig function. On the calculator, you can find the inverse trig functions by pressing 2 nd and then the trig function. On the calculator, you can find the inverse trig functions by pressing 2 nd and then the trig function.

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Let’s Look at an Example Given the following triangle, solve for x. Given the following triangle, solve for x. 200 cm 62 cm 90° x

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Step 2 Label the sides opposite, adjacent, or hypotenuse from angle x. 200 cm 62 cm 90° x OPPOSITE HYPOTENUSE

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Step 3 Since we have the opposite and the hypotenuse, we need to use SINE. 200 cm 62 cm 90° x HYPOTENUSE OPPOSITE

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Step 4 Substitute in the given information into the equation. 200 cm 62 cm 90° x HYPOTENUSE OPPOSITE Sin x =

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Step 5 To solve for x, we need to take the inverse of sine on both sides. 200 cm 62 cm 90° x HYPOTENUSE OPPOSITE Sin x = Sin -1 (sin x) = Sin -1

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Step 6 Now just type in the x= on your calculator. Sin x = Sin -1 (sin x) = Sin -1 X = Sin -1 X = 18°

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Now It’s Your Turn! Use what you’ve just reviewed to help you answer the following questions. Use what you’ve just reviewed to help you answer the following questions. Submit all of your work to your teacher after completing the tutorial. Submit all of your work to your teacher after completing the tutorial. Don’t be afraid to go back through the slides if you get stuck. Don’t be afraid to go back through the slides if you get stuck. GOOD LUCK! GOOD LUCK!

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Problem #1 Complete the following ratios. Complete the following ratios. C A B 90° 6 cm 8 cm 10 cm Sin A = Sin B = Cos A=Cos B= Tan A=Tan B=

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Problem #2 Solve for x and y. Solve for x and y. 90° 55° x 40 ft y

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Problem #3 Solve for angles A and B. Solve for angles A and B. 90° A 5 in B C 7 in

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