Question Video: Solving Reciprocal Trigonometric Equations | Nagwa Question Video: Solving Reciprocal Trigonometric Equations | Nagwa

Question Video: Solving Reciprocal Trigonometric Equations Mathematics • First Year of Secondary School

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Find the set of values satisfying 2 sin 𝜃 csc 𝜃 + sec 𝜃 cot 𝜃 = 0 given that 0° ≤ 𝜃 ≤ 360°.

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Video Transcript

Find the set of values satisfying two sin 𝜃 csc 𝜃 plus sec 𝜃 cot 𝜃 equals zero given that 𝜃 is greater than or equal to zero degrees and less than or equal to 360 degrees.

We will begin this question by recalling the definition of the cosecant, secant, and cotangent functions. csc of angle 𝜃 is equal to one over sin 𝜃. The sec of angle 𝜃 is equal to one over cos 𝜃. And the cot of 𝜃 is equal to one over the tan of 𝜃. We also recall that since tan 𝜃 is equal to sin 𝜃 over cos 𝜃, the cot of 𝜃 is equal to cos 𝜃 over sin 𝜃. Substituting these identities into our equation, we have two sin 𝜃 multiplied by one over sin 𝜃 plus one over cos 𝜃 multiplied by cos 𝜃 over sin 𝜃 is equal to zero. The first term on the left-hand side simplifies to two, and the second term simplifies to one over sin 𝜃.

Subtracting two from both sides, we have one over sin 𝜃 equals negative two. This means that the csc of 𝜃 equals negative two, which in turn tells us that sin of angle 𝜃 is equal to negative one-half. We can solve this equation by firstly sketching a CAST diagram. As the sign of angle 𝜃 is negative, there will be solutions in the third and fourth quadrants.

From our knowledge of special angles, we know that the sin of 30 degrees is equal to a half. We can then use our knowledge of the symmetry of the sine function to calculate the solutions in the third and fourth quadrants. In the third quadrant, we have 𝜃 is equal to 180 degrees plus 30 degrees. This is equal to 210 degrees. In the fourth quadrant, we have 𝜃 is equal to 360 degrees minus 30 degrees. This is equal to 330 degrees. The set of values that satisfy two sin 𝜃 csc 𝜃 plus sec 𝜃 cot 𝜃 equals zero where 𝜃 lies between zero and 360 degrees inclusive are 210 and 330 degrees.

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