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Question Video: Using Reciprocal Identities to Simplify a Trigonometric Expression Mathematics • 10th Grade

Find the value of (8/sin ๐œƒ) ร— (โˆ’5/csc ๐œƒ).

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Video Transcript

Find the value of eight over sin ๐œƒ times negative five over csc ๐œƒ.

In this expression, we have a trig function and a reciprocal trig function. One strategy for evaluating a trigonometric expression is to write it in terms of sine and cosine functions. The first term in this expression is already in terms of sine and cosine. And to rewrite csc ๐œƒ in terms of sine or cosine, we recall that csc ๐œƒ is the reciprocal of sin ๐œƒ and that csc ๐œƒ equals one over sin ๐œƒ.

Therefore, in the denominator of this second term, we replace csc ๐œƒ with one over sin ๐œƒ. If we think about negative five over one over sin ๐œƒ, this is negative five divided by one over sin ๐œƒ. And dividing by a fraction is the same as multiplying by the reciprocal of that fraction. Therefore, we can rewrite negative five over one over sin ๐œƒ as negative five times sin of ๐œƒ. In this case, we have a sin ๐œƒ in the denominator and a sin ๐œƒ in the numerator, which cancels out, leaving us with eight times negative five, which is negative 40.

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