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Math  /  Trigonometry

QuestionUse the appropriate angle sum or difference formula to find the exact value of sin⁡(−435∘)\sin \left(-435^{\circ}\right).

Studdy Solution

STEP 1

1. We are given the angle −435∘-435^\circ.
2. We need to find the exact value of sin⁡(−435∘)\sin(-435^\circ).
3. We will use angle sum or difference formulas and trigonometric identities.

STEP 2

1. Simplify the angle −435∘-435^\circ to an equivalent angle within the standard range [0∘,360∘)[0^\circ, 360^\circ).
2. Use the angle sum or difference formula for sine if necessary.
3. Calculate the sine of the simplified angle.

STEP 3

Simplify −435∘-435^\circ by adding 360∘360^\circ until the angle is within the range [0∘,360∘)[0^\circ, 360^\circ):
−435∘+360∘=−75∘ -435^\circ + 360^\circ = -75^\circ
Since −75∘-75^\circ is still negative, add 360∘360^\circ again:
−75∘+360∘=285∘ -75^\circ + 360^\circ = 285^\circ
The equivalent angle in the standard range is 285∘285^\circ.

STEP 4

Recognize that 285∘285^\circ can be expressed as 360∘−75∘360^\circ - 75^\circ.
Use the identity sin⁡(360∘−θ)=−sin⁡(θ)\sin(360^\circ - \theta) = -\sin(\theta).
sin⁡(285∘)=sin⁡(360∘−75∘)=−sin⁡(75∘) \sin(285^\circ) = \sin(360^\circ - 75^\circ) = -\sin(75^\circ)

STEP 5

Calculate sin⁡(75∘)\sin(75^\circ) using the angle sum formula:
sin⁡(75∘)=sin⁡(45∘+30∘) \sin(75^\circ) = \sin(45^\circ + 30^\circ)
Use the angle sum formula:
sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B \sin(A + B) = \sin A \cos B + \cos A \sin B
Substitute A=45∘A = 45^\circ and B=30∘B = 30^\circ:
sin⁡(75∘)=sin⁡(45∘)cos⁡(30∘)+cos⁡(45∘)sin⁡(30∘) \sin(75^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ)
Using known values:
sin⁡(45∘)=22\sin(45^\circ) = \frac{\sqrt{2}}{2}, cos⁡(30∘)=32\cos(30^\circ) = \frac{\sqrt{3}}{2}, cos⁡(45∘)=22\cos(45^\circ) = \frac{\sqrt{2}}{2}, sin⁡(30∘)=12\sin(30^\circ) = \frac{1}{2}
Substitute these values:
sin⁡(75∘)=22⋅32+22⋅12 \sin(75^\circ) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}
sin⁡(75∘)=64+24 \sin(75^\circ) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}
sin⁡(75∘)=6+24 \sin(75^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4}
Thus:
sin⁡(285∘)=−sin⁡(75∘)=−6+24 \sin(285^\circ) = -\sin(75^\circ) = -\frac{\sqrt{6} + \sqrt{2}}{4}
The exact value of sin⁡(−435∘)\sin(-435^\circ) is:
−6+24 \boxed{-\frac{\sqrt{6} + \sqrt{2}}{4}}

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