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**Unformatted text preview: **nd all angles θ for which sin(θ) = 3 . 2. Find all real numbers t for which tan(t) = −2
3. Solve sec(x) = − 5 for x.
3
Solution.
1
1. If sin(θ) = 3 , then the terminal side of θ, when plotted in standard position, intersects the
Unit Circle at y = 1 . Geometrically, we see that this happens at two places: in Quadrant I
3
and Quadrant II.
y 1 1
3 y 1 α
1 α
x 1
3 1 x 1
The quest now is to ﬁnd the measures of these angles. Since 3 isn’t the sine of any of the
‘common angles’ discussed earlier, we are forced to use the inverse trigonometric functions,
in this case the arcsine function, to express our answers. By deﬁnition, the real number
t = arcsin 1 satisﬁes 0 < t < π with sin(t) = 1 , so we know our solutions have a reference
3
2
3
angle of α = arcsin 1 radians. The solutions in Quadrant I are all coterminal with α and
3
1
so our solution here is θ = α + 2πk = arcsin 3 + 2πk for integers k . Turning our attention
to Quadrant II, one angle with a reference angle of α is π − α. Hen...

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