Question Video: Using the Pythagorean Identities to Evaluate the Sine Function given the Cosine Function and the Quadrant of an Angle Mathematics

Find the value of sin 𝜃 given cos 𝜃 = −21/29 where 90° < 𝜃 < 180°.

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Video Transcript

Find the value of sin 𝜃 given cos of 𝜃 is equal to negative 21 over 29, where 𝜃 is greater than 90 degrees and less than 180 degrees.

In order to answer this question, we’ll use the Pythagorean identity sin squared 𝜃 plus cos squared 𝜃 is equal to one. We are told that 𝜃 lies between 90 and 180 degrees, so it is worth considering our CAST diagram. The angle lies in the second quadrant, which means that the sin of angle 𝜃 must be positive. The cos of angle 𝜃 and tan of angle 𝜃 must be negative. This ties in with the fact that we are told that the cos of 𝜃 is equal to negative 21 over 29. It also helps us in that we know the answer for sin 𝜃 must be positive.

We can substitute the value of cos 𝜃 into the Pythagorean identity. Squaring negative 21 over 29 gives us 441 over 841. We can then subtract this from both sides of our equation such that sin squared 𝜃 is equal to one minus 441 over 841. The right-hand side simplifies to 400 over 841. We can then square root both sides of our equation so that sin of 𝜃 is equal to positive or negative the square root of 400 over 841.

Square rooting the numerator gives us 20 and the denominator 29. As sin of 𝜃 must be positive, if the cos of 𝜃 is negative 21 over 29 and 𝜃 lies between 90 and 180 degrees, then the sin of 𝜃 is equal to 20 over 29.

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