Video: Evaluating Trigonometric Expressions Using Reciprocal Identities

Find the value of 2 sin πœƒ csc πœƒ +11 cos πœƒ sec πœƒ + 7 tan πœƒ cot πœƒ.

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Video Transcript

Find the value of two sin πœƒ csc πœƒ plus 11 cos πœƒ sec πœƒ plus seven tan πœƒ cot πœƒ.

To solve this problem, we need to recognize reciprocal identities. First, let’s think about ways we can rewrite cosecant, secant, and cotangent. Csc is equal to one over sin πœƒ. Sec πœƒ is one over cos πœƒ. And cot πœƒ is equal to one over tan πœƒ. What does this mean? Well, one over sin πœƒ is the reciprocal of sin πœƒ. And that means cosecant is the reciprocal of sine. It means that sec πœƒ is the reciprocal of cos πœƒ, and that cotangent is the reciprocal of tangent.

If we multiply sin πœƒ by its reciprocal, one over sin πœƒ, we’ll get sin πœƒ over sin πœƒ, which is one. Anything multiplied by its reciprocal is one. Two times sin πœƒ times csc πœƒ is equal to two times one. The same thing is true for cos πœƒ times sec πœƒ. Cos πœƒ times its reciprocal equals one. And so, the second term will be 11 times one. Tan πœƒ times its reciprocal equals one. And so, our third term will be seven times one. We can simplify this expression to, say, two plus 11 plus seven, which equals 20.

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