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To find the values of [tex]$\sin 15^{\circ}$[/tex], [tex]$\cos 15^{\circ}$[/tex], and [tex]$\tan 15^{\circ}$[/tex], we can use the half-angle formulas. Given that [tex]$\cos 30^{\circ} = \frac{\sqrt{3}}{2}$[/tex], we'll follow these steps:
### Step 1: Calculate [tex]$\sin 15^{\circ}$[/tex]
We use the half-angle formula for sine, which states:
[tex]\[ \sin \left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos \theta}{2}} \][/tex]
Here, [tex]$\theta = 30^{\circ}$[/tex], so we have:
[tex]\[ \sin 15^{\circ} = \sin \left(\frac{30^{\circ}}{2}\right) = \sqrt{\frac{1 - \cos 30^{\circ}}{2}} \][/tex]
Substituting the given value of [tex]$\cos 30^{\circ}$[/tex]:
[tex]\[ \sin 15^{\circ} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \][/tex]
Calculating inside the square root:
[tex]\[ \sin 15^{\circ} = \sqrt{\frac{2/2 - \sqrt{3}/2}{2}} = \sqrt{\frac{2 - \sqrt{3}}{4}} = \sqrt{\frac{2 - \sqrt{3}}{4}} \][/tex]
Taking the square root:
[tex]\[ \sin 15^{\circ} \approx 0.2588 \][/tex]
### Step 2: Calculate [tex]$\cos 15^{\circ}$[/tex]
We use the half-angle formula for cosine, which states:
[tex]\[ \cos \left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos \theta}{2}} \][/tex]
Again, with [tex]$\theta = 30^{\circ}$[/tex]:
[tex]\[ \cos 15^{\circ} = \cos \left(\frac{30^{\circ}}{2}\right) = \sqrt{\frac{1 + \cos 30^{\circ}}{2}} \][/tex]
Substituting the given value of [tex]$\cos 30^{\circ}$[/tex]:
[tex]\[ \cos 15^{\circ} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} \][/tex]
Calculating inside the square root:
[tex]\[ \cos 15^{\circ} = \sqrt{\frac{2/2 + \sqrt{3}/2}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}} \][/tex]
Taking the square root:
[tex]\[ \cos 15^{\circ} \approx 0.9659 \][/tex]
### Step 3: Calculate [tex]$\tan 15^{\circ}$[/tex]
The tangent of an angle is the ratio of the sine of that angle to the cosine of that angle:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
Thus,
[tex]\[ \tan 15^{\circ} = \frac{\sin 15^{\circ}}{\cos 15^{\circ}} \][/tex]
Substituting the approximated values:
[tex]\[ \tan 15^{\circ} = \frac{0.2588}{0.9659} \][/tex]
Calculating the ratio:
[tex]\[ \tan 15^{\circ} \approx 0.2679 \][/tex]
### Summary
Thus, the values are approximately:
[tex]\[ \sin 15^{\circ} \approx 0.2588, \quad \cos 15^{\circ} \approx 0.9659, \quad \tan 15^{\circ} \approx 0.2679 \][/tex]
### Step 1: Calculate [tex]$\sin 15^{\circ}$[/tex]
We use the half-angle formula for sine, which states:
[tex]\[ \sin \left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos \theta}{2}} \][/tex]
Here, [tex]$\theta = 30^{\circ}$[/tex], so we have:
[tex]\[ \sin 15^{\circ} = \sin \left(\frac{30^{\circ}}{2}\right) = \sqrt{\frac{1 - \cos 30^{\circ}}{2}} \][/tex]
Substituting the given value of [tex]$\cos 30^{\circ}$[/tex]:
[tex]\[ \sin 15^{\circ} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} \][/tex]
Calculating inside the square root:
[tex]\[ \sin 15^{\circ} = \sqrt{\frac{2/2 - \sqrt{3}/2}{2}} = \sqrt{\frac{2 - \sqrt{3}}{4}} = \sqrt{\frac{2 - \sqrt{3}}{4}} \][/tex]
Taking the square root:
[tex]\[ \sin 15^{\circ} \approx 0.2588 \][/tex]
### Step 2: Calculate [tex]$\cos 15^{\circ}$[/tex]
We use the half-angle formula for cosine, which states:
[tex]\[ \cos \left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos \theta}{2}} \][/tex]
Again, with [tex]$\theta = 30^{\circ}$[/tex]:
[tex]\[ \cos 15^{\circ} = \cos \left(\frac{30^{\circ}}{2}\right) = \sqrt{\frac{1 + \cos 30^{\circ}}{2}} \][/tex]
Substituting the given value of [tex]$\cos 30^{\circ}$[/tex]:
[tex]\[ \cos 15^{\circ} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} \][/tex]
Calculating inside the square root:
[tex]\[ \cos 15^{\circ} = \sqrt{\frac{2/2 + \sqrt{3}/2}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}} \][/tex]
Taking the square root:
[tex]\[ \cos 15^{\circ} \approx 0.9659 \][/tex]
### Step 3: Calculate [tex]$\tan 15^{\circ}$[/tex]
The tangent of an angle is the ratio of the sine of that angle to the cosine of that angle:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
Thus,
[tex]\[ \tan 15^{\circ} = \frac{\sin 15^{\circ}}{\cos 15^{\circ}} \][/tex]
Substituting the approximated values:
[tex]\[ \tan 15^{\circ} = \frac{0.2588}{0.9659} \][/tex]
Calculating the ratio:
[tex]\[ \tan 15^{\circ} \approx 0.2679 \][/tex]
### Summary
Thus, the values are approximately:
[tex]\[ \sin 15^{\circ} \approx 0.2588, \quad \cos 15^{\circ} \approx 0.9659, \quad \tan 15^{\circ} \approx 0.2679 \][/tex]
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