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Difficulty: Very hardTrigonometric Graphs and Simple Equations

Find the sum of all values of xx (in degrees) in the interval 0∘≤x≤180∘0^\circ \le x \le 180^\circ that satisfy the trigonometric equation cos⁡(3x−45∘)=−22\cos(3x - 45^\circ) = -\frac{\sqrt{2}}{2}.

Answer: 330 degrees

Answer

The sum of all values of xx satisfying the equation in the domain 0∘≤x≤180∘0^\circ \le x \le 180^\circ is 330.
Transforming the domain 0∘≤x≤180∘0^\circ \le x \le 180^\circ gives −45∘≤3x−45∘≤495∘-45^\circ \le 3x - 45^\circ \le 495^\circ. The angles within this range where the cosine value equals −22-\frac{\sqrt{2}}{2} are 135∘135^\circ, 225∘225^\circ, and 495∘495^\circ. Solving 3x−45∘3x - 45^\circ for each of these angles gives x=60∘x = 60^\circ, 90∘90^\circ, and 180∘180^\circ. Summing these three roots yields 330∘330^\circ.

Step-by-Step Solution

1
Determine the interval of the transformed angle θ=3x−45∘\theta = 3x - 45^\circ.
−45∘≤3x−45∘≤495∘-45^\circ \le 3x - 45^\circ \le 495^\circ
Applying the linear transformation 3x−45∘3x - 45^\circ to the given domain 0∘≤x≤180∘0^\circ \le x \le 180^\circ establishes the exact boundaries for the argument of the cosine function.
2
Find all values of θ\theta within [−45∘,495∘][-45^\circ, 495^\circ] satisfying cos⁡θ=−22\cos \theta = -\frac{\sqrt{2}}{2}.
θ∈{135∘,225∘,495∘}\theta \in \{135^\circ, 225^\circ, 495^\circ\}
Cosine is negative in Quadrants II and III. The reference angle is 45∘45^\circ, giving base solutions 135∘135^\circ and 225∘225^\circ. Adding 360∘360^\circ to 135∘135^\circ gives 495∘495^\circ, which lies exactly on the upper boundary.
3
Solve for xx by setting 3x−45∘3x - 45^\circ equal to each valid θ\theta.
x∈{60∘,90∘,180∘}x \in \{60^\circ, 90^\circ, 180^\circ\}
Isolating xx yields x=θ+45∘3x = \frac{\theta + 45^\circ}{3}. All three resulting values lie within [0∘,180∘][0^\circ, 180^\circ].
4
Sum the valid solution values.
60^\circ + 90^\circ + 180^\circ = 330^\circ
The problem asks specifically for the sum of all solution angles in degrees.

Key Concept

Solving multi-angle trigonometric equations with phase shifts across a specified domain
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Topics

Trigonometric Graphs and Simple Equations

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