solve---**Question 6** **Question Stem:** For each of the following angles, state the other angle between −180° and 180° that gives the same value for: **Sub-questions / Conditions:** i. sin θ ii. cos θ **Angles to consider (for which to find the other angle):** a. 70° b. 40° c. 110° d. 155° e. −15° f. −68° g. −100° h. −145°

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Welcome to solving angle equivalence problems. We need to find other angles between negative 180 degrees and positive 180 degrees that give the same sine and cosine values. Trigonometric functions are periodic, meaning different angles can produce identical values. On the unit circle, we can see how angles in different quadrants relate to each other. For example, 70 degrees and 110 degrees both have the same sine value due to symmetry. We'll analyze eight specific angles: the positive angles 70, 40, 110, and 155 degrees, and the negative angles negative 15, negative 68, negative 100, and negative 145 degrees. Now let's examine the sine function's symmetry properties. The key property is that sine of theta equals sine of 180 degrees minus theta. This creates symmetry about the y-axis in the upper half of the unit circle. For example, sine of 30 degrees equals sine of 150 degrees, both equal to 0.5. We can see this visually as both points have the same y-coordinate, which represents the sine value. The sine function also has odd function properties, where sine of theta equals negative sine of negative theta. These symmetries help us find equivalent angles that produce the same sine values. Now let's examine the cosine function's symmetry properties. The key property is that cosine of theta equals cosine of negative theta, making cosine an even function. This creates symmetry about the x-axis. For example, cosine of 60 degrees equals cosine of negative 60 degrees, both equal to 0.5. We can see this visually as both points have the same x-coordinate, which represents the cosine value. The cosine function also has the property that cosine of theta equals negative cosine of 180 degrees minus theta, showing the relationship between supplementary angles. These symmetries help us find equivalent angles that produce the same cosine values. Now let's solve the positive angles systematically. For 70 degrees, the sine equivalent is 180 minus 70, which equals 110 degrees. The cosine equivalent is negative 70 degrees. For 40 degrees, the sine equivalent is 180 minus 40, which equals 140 degrees. The cosine equivalent is negative 40 degrees. For 110 degrees, the sine equivalent is 180 minus 110, which equals 70 degrees. The cosine equivalent is negative 110 degrees. For 155 degrees, the sine equivalent is 180 minus 155, which equals 25 degrees. The cosine equivalent is negative 155 degrees. We use the relationships sine of theta equals sine of 180 degrees minus theta, and cosine of theta equals cosine of negative theta. Now let's solve the negative angles while ensuring results stay within negative 180 to positive 180 degrees. For negative 15 degrees, the sine equivalent is negative 165 degrees, and the cosine equivalent is positive 15 degrees. For negative 68 degrees, the sine equivalent is negative 112 degrees, and the cosine equivalent is positive 68 degrees. For negative 100 degrees, the sine equivalent is negative 80 degrees, and the cosine equivalent is positive 100 degrees. For negative 145 degrees, the sine equivalent is negative 35 degrees, and the cosine equivalent is positive 145 degrees. We use the properties that sine of negative theta equals negative sine of theta, cosine of negative theta equals cosine of theta, and the supplementary angle relationships to find equivalent angles within our required range.