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Find sin π΅, given π΄π΅πΆ is a right-angled triangle at πΆ, where π΄π΅ equals 17 centimeters and π΅πΆ equals 15 centimeters.
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So we have a right-angled triangle, in which weβve been given the lengths of two of its sides.
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Weβre asked to find the value of sin π΅, which is a trigonometric ratio in reference to the angle π΅.
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Letβs recall the definition of sine.
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In a right-angled triangle, the sine of an angle, in this case angle π΅, is equal to the ratio of the opposite side divided by the hypotenuse.
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Letβs look at which side is which for the triangle in this question.
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Side π΄π΅ is the hypotenuse because itβs opposite the right angle.
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With respect to angle π΅, side π΅πΆ is the adjacent because itβs between angle π΅ and the right angle.
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Again, with respect to angle π΅, side π΄πΆ is the opposite.
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So this means that in this triangle, the ratio of sin π΅ is going to be found by dividing the length of π΄πΆ by the length of π΄π΅.
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Now, we know the length of π΄π΅; itβs given in the diagram 17 centimeters, but we donβt know the length of π΄πΆ.
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So in order to calculate this sine ratio, we need a method of finding the length of π΄πΆ.
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Letβs think about what else we know about right-angled triangles.
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In this triangle, we know the length of two of the sides and we want to calculate the length of third, which means this is the perfect setting to apply the Pythagorean theorem.
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Remember the Pythagorean theorem tells us that in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
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In this triangle, this means that π΄πΆ squared plus 15 squared is equal to 17 squared.
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Now, this is an equation that we can solve in order to find the length of π΄πΆ.
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Evaluating 15 squared and 17 squared gives π΄πΆ squared plus 225 is equal to 289.
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Next, we subtract 225 from both sides and this gives π΄πΆ squared is equal to 64.
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Finally, we square root both sides of this equation π΄πΆ is equal to the square root of 64, which is eight.
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So now, we know the length of the third side of the triangle.
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You may also have spotted this quite quickly, if youβre familiar with your Pythagorean triples because eight, 15, 17 is an example of one.
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In any case, we now know the length of π΄πΆ.
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So letβs return to the focus of this question, which was to calculate the trigonometric ratio sin of π΅.
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We found that it was equal to π΄πΆ over 17.
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We now know that π΄πΆ is equal to eight.
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Therefore, sin of angle π΅ is equal to eight over 17.