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MFM2P Triangles: The Sine Ratio Name: _________________________ Date: __________________________ Look at the two triangles below. If we know ∆ABC ~ ∆DEF, explore the ratio of opposite with our hypotenuse angle of interest of both triangles. € In ∆ABC, our opposite side length is ________, the hypotenuse length is ________ and the ratio is ________. In ∆DEF, our opposite side length is ________, the hypotenuse length is ________ and the ratio is ________. What can you say about their ratios? _________________________________________________________________________ We use the ratios of side lengths for triangles so often that we developed a tool to find side lengths of triangles more easily. Just like a fridge is a tool to keep food cold and a hammer is a tool to hit nails, the sine ratio is a tool for angles of a right-‐angle triangle. Going back to the example above, what were the ratios of both triangles? _____________________________ On your calculator, calculate sin 30°. What do you get? ______________________ We developed the tool called the sine ratio that says: SIN A = length of opposite side length of hypotenuse This formula allows us to now find a side length without using the Pythagorean theorem! Woot! So what information do we need? • ANGLE OF INTEREST And one of the following: • OPPOSITE SIDE • HYPOTENUSE Example: Calculate the length of x. sin A = length of opposite side length of hypotenuse sin 30° = x 7cm (7cm)(sin 30°) = x (7cm)(0.5) = x 3.5cm = x The length of x is 3.5cm. Example: Calculate the length of the hypotenuse, y. opposite sin A = hypotenuse sin 57° = € 6.5 sin57 o y = € 6.5 y = 6.5 0.838 € = 7.75 € The length of the hypotenuse y is 7.75 units. What if we have two sides but no angles? What about working backwards??? We can use the inverse of sine, or the sin-‐1(A) button on your calculator, to find our angle. Example: Find the angle of interest for this triangle. opposite sin θ = hypotenuse sin θ = € 12 15 sin θ = 0.8 € θ = sin−1 (0.8) € θ = 53.1o € € *-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐*-‐* In-Class Work 1. Evaluate and round to 3 decimal places (if necessary): a) sin 30° b) sin 45° c) sin 65° d) sin 135° 2. Label the hypotenuse of this triangle. Then, label the opposite side of the angle of interest with a number smaller than you chose for the hypotenuse. Calculate the angle (using sin-‐1) for the triangle you chose. 3. Determine the measure of the angle, to the nearest degree. a) sin W = 0.8192 b) sin X = 0.9063 c) sin Y = 0.4226 4. Determine the measure of each angle using the sine ratio. a) b) d) sin J = 0.2533
