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9.4 Using Trigonometry to Find Missing Sides of Right Triangles Introduction: What method can we use to find x in the triangle below? Pythagorean Theorem • Can we use the same method to find x in the following triangle? Trigonometry can help us to find measures of sides and angles of right triangles that we were unable to find when our only tool was the Pythagorean theorem. Remember that we are using Trigonometry at this point to find missing sides of Right Triangles What other angle (other than the right angle) is given in the picture below? Reference Angle (starting angle) Using the 34º angle as your reference angle, would “x” be the opposite leg, the adjacent leg, or the hypotenuse? Label each side with a O, A or H. H O A Remember the 3 Trig Ratios: Sine (𝜃) = 𝑜𝑝𝑝 ℎ𝑦𝑝 Sine (𝜃) = Cos (𝜃) = 𝑜𝑝𝑝 ℎ𝑦𝑝 “(𝜃)” Reference Angle 𝑎𝑑𝑗 𝐻𝑦𝑝 Tan (𝜃) = 𝑜𝑝𝑝 𝑎𝑑𝑗 Sine (𝜃) = 𝑜𝑝𝑝 ℎ𝑦𝑝 Remember that “(𝜃)” stands for the reference angle. What is “(𝜃)” ? What is the length of the opposite leg? What is the length of the hypotenuse? 𝑥 12 Sine (34°) = X = 12sin(34°) ≈12(0.5592) ≈ 6.71 So x ≈ 6.71 Let us look once again at the following triangle: • What is the measure of the other acute angle? • Now repeat the same process that we followed in the previous slide. • Use the new acute angle as the reference angle 𝑥 ???? ( ? °) = 12 X = 12 ??? ( ? °) ≈12( ??) ≈ ???? So x ≈ ? ? ? 56° Cos (56°) = 𝑥 12 X = 12cos(56°) ≈12(0.5592) ≈6.71 So x ≈ 6.71 Compare the value from the other angle of 34°. Notice that we obtain the same value of “x” regardless of which reference angle we choose to use. You Try: Find x in the triangle. Round answer to the nearest hundred #1 18.4 x ≈ ___________ You Try: Find x in the triangle. Round answer to the nearest hundred #2 13.1 x ≈ ___________ You Try: Find x in the triangle. Round answer to the nearest hundred #3 7.5 x ≈ ___________ The End