Question Video: Solving a Trigonometric Equation Involving the Secant Function | Nagwa Question Video: Solving a Trigonometric Equation Involving the Secant Function | Nagwa

Question Video: Solving a Trigonometric Equation Involving the Secant Function Mathematics • First Year of Secondary School

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Find the solution set of 𝜃 that satisfies sec (𝜃) = −√2 given that 0° ≤ 𝜃 < 360°.

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Video Transcript

Find the solution set of 𝜃 that satisfies sec 𝜃 is equal to negative root two given that 𝜃 is greater than or equal to zero degrees and less than 360 degrees.

In order to answer this question, we begin by recalling that the secant function is the reciprocal of the cosine function, such that sec 𝜃 is equal to one over cos 𝜃. This means that we can rewrite the equation as one over cos is equal to negative root two. And in turn, cos 𝜃 is equal to negative one over root two. We can solve this equation using inverse trigonometric functions together with our knowledge of the CAST diagram and special angles.

Since the value of cos 𝜃 is negative and lies between zero and negative one, we know we will have solutions in the second and third quadrants. At this stage, we may recall that the cos of 45 degrees is equal to one over root two. We can use this fact together with the symmetry of the cosine function in the CAST diagram to find the solutions of our equation in the second and third quadrants.

The values of 𝜃 that satisfy the equation cos 𝜃 is equal to negative one over root two are 𝜃 is equal to 180 degrees minus 45 degrees and 𝜃 is equal to 180 degrees plus 45 degrees. This gives us our two solutions of 135 degrees and 225 degrees.

It is worth noting we could also have solved the equation by taking the inverse cosine of both sides, such that 𝜃 is equal to the inverse cos of negative one over root two. Ensuring that our calculator is in degree mode, typing in the right-hand side would have given us our first solution of 135 degrees. We could then have found the second solution by subtracting this value from 360 degrees, giving us our second solution of 225 degrees.

Either way, the solution set of 𝜃 that satisfies sec 𝜃 is equal to negative root two where 𝜃 is greater than or equal to zero and less than 360 degrees is 135 degrees and 225 degrees.

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