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Find sin 𝐶 multiplied by cos 𝐶 given that 𝐴𝐵𝐶 is a right triangle at 𝐵, where 𝐴𝐵 equals eight centimeters and 𝐴𝐶 equals 17 centimeters.
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In order to answer this question, we will use our knowledge of trigonometry in right triangles.
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The trigonometric ratios tell us that the sin of angle 𝜃 is equal to the opposite over the hypotenuse.
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The cos of angle 𝜃 is the adjacent over the hypotenuse.
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And the tan of angle 𝜃 is equal to the opposite over the adjacent.
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One way of remembering this is using the acronym SOHCAHTOA.
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In this question, the angle 𝜃 is at point 𝐶 of our triangle.
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We know that the hypotenuse of any right triangle is opposite the right angle and is the longest side.
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In this case, this is the length 𝐴𝐶, which is equal to 17 centimeters.
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This side 𝐴𝐵 is opposite angle 𝐶, and this has a length of eight centimeters.
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The side 𝐵𝐶 is the adjacent side of our triangle.
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In order to calculate the length of this side of our triangle, we will use the Pythagorean theorem, which states that 𝑥 squared plus 𝑦 squared is equal to 𝑧 squared, where 𝑧 is the length of the hypotenuse or longest side of the triangle.
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In this question, 𝐵𝐶 squared plus eight squared is equal to 17 squared.
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Eight squared is equal to 64, and 17 squared is 289.
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We can then subtract 64 from both sides of this equation.
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𝐵𝐶 squared is therefore equal to 225.
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And taking the square root of both sides gives us 𝐵𝐶 is equal to 15.
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As the side length must be positive, we have 𝐵𝐶 is equal to 15 centimeters, noting that this triangle is an eight-15-17 Pythagorean triple.
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We can now work out the values of sin 𝐶 and cos 𝐶.
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As sin 𝜃 is equal to the opposite over the hypotenuse, sin 𝐶 is equal to eight over 17.
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In the same way, as the cos of angle 𝜃 is the adjacent over the hypotenuse, the cos of angle 𝐶 is 15 over 17.
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To calculate the product of two fractions, we simply multiply the numerators and then separately multiply the denominators.
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Eight multiplied by 15 is 120, and 17 multiplied by 17, or 17 squared, is 289.
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In the right triangle 𝐴𝐵𝐶 as shown, sin 𝐶 multiplied by cos 𝐶 is 120 over 289.