Questions: 4. sin(90°)= 5. cos(225°)= 6. sin(210°)=

4. sin(90°)= 5. cos(225°)= 6. sin(210°)=
Transcript text: 4. $\sin \left(90^{\circ}\right)=$ 5. $\cos \left(225^{\circ}\right)=$ 6. $\sin \left(210^{\circ}\right)=$ Remark: Your answer cannot contain trigonometric functions; it must be a fraction (not a decimal) and it may contain square roots (e.g., sqrt(2)).
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Solution

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Solution Steps

To solve these trigonometric problems, we will use the unit circle and trigonometric identities. The sine and cosine functions have specific values at key angles, which can be expressed as fractions involving square roots.

  1. For \(\sin(90^\circ)\), we know from the unit circle that the sine of 90 degrees is 1.
  2. For \(\cos(225^\circ)\), we use the fact that 225 degrees is in the third quadrant where cosine is negative. The reference angle is 45 degrees, and \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\), so \(\cos(225^\circ) = -\frac{\sqrt{2}}{2}\).
  3. For \(\sin(210^\circ)\), 210 degrees is in the third quadrant where sine is negative. The reference angle is 30 degrees, and \(\sin(30^\circ) = \frac{1}{2}\), so \(\sin(210^\circ) = -\frac{1}{2}\).
Step 1: Calculate \(\sin(90^\circ)\)

From the unit circle, we know that: \[ \sin(90^\circ) = 1 \]

Step 2: Calculate \(\cos(225^\circ)\)

The angle \(225^\circ\) is in the third quadrant, where cosine values are negative. The reference angle is \(45^\circ\), and we have: \[ \cos(225^\circ) = -\cos(45^\circ) = -\frac{\sqrt{2}}{2} \] The decimal approximation is: \[ \cos(225^\circ) \approx -0.7071 \]

Step 3: Calculate \(\sin(210^\circ)\)

The angle \(210^\circ\) is also in the third quadrant, where sine values are negative. The reference angle is \(30^\circ\), and we have: \[ \sin(210^\circ) = -\sin(30^\circ) = -\frac{1}{2} \]

Final Answer

Thus, the results are: \[ \sin(90^\circ) = 1, \quad \cos(225^\circ) \approx -0.7071, \quad \sin(210^\circ) = -\frac{1}{2} \] The final answers are: \[ \boxed{\sin(90^\circ) = 1}, \quad \boxed{\cos(225^\circ) \approx -0.7071}, \quad \boxed{\sin(210^\circ) = -\frac{1}{2}} \]

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