122.) $$ \sin 107 $$ 128. $$ \cos ^{2} 253+\sin ^{2} 557 $$

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**122.) Compute $$ \sin 107^\circ $$**

1. Recognize that $$107^\circ$$ is in the second quadrant where the sine is positive.
2. Use the identity 
   $$
   \sin (180^\circ - x) = \sin x.
   $$
3. Write
   $$
   \sin 107^\circ = \sin (180^\circ - 107^\circ) = \sin 73^\circ.
   $$
4. (Optional) If a numerical approximation is desired, then
   $$
   \sin 73^\circ \approx 0.9563.
   $$

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**128.) Compute $$ \cos^{2} 253^\circ + \sin^{2} 557^\circ $$**

1. **Reduce the angles:**

- For $$ \cos 253^\circ $$:
     - Write $$253^\circ$$ as $$180^\circ + 73^\circ$$. Since cosine has the property
       $$
       \cos (180^\circ + x) = -\cos x,
       $$
       we have
       $$
       \cos 253^\circ = -\cos 73^\circ.
       $$
     - Therefore,
       $$
       \cos^2 253^\circ = (\cos 73^\circ)^2.
       $$

- For $$ \sin 557^\circ $$:
     - Reduce by subtracting $$360^\circ$$: 
       $$
       557^\circ - 360^\circ = 197^\circ.
       $$
     - Write $$197^\circ$$ as $$180^\circ + 17^\circ$$. Since sine has the property
       $$
       \sin (180^\circ + x) = -\sin x,
       $$
       we find
       $$
       \sin 197^\circ = -\sin 17^\circ.
       $$
     - Thus,
       $$
       \sin^2 557^\circ = (\sin 17^\circ)^2.
       $$

2. **Express in a common form:**

Notice that using the co-function identity,
   $$
   \sin 17^\circ = \cos (90^\circ - 17^\circ) = \cos 73^\circ.
   $$

3. **Combine the results:**

Substitute the above into the given expression:
   $$
   \cos^{2} 253^\circ + \sin^{2} 557^\circ = \cos^2 73^\circ + \sin^2 17^\circ = \cos^2 73^\circ + \cos^2 73^\circ = 2\cos^2 73^\circ.
   $$

4. **Optional alternative form:**

Using the double-angle identity,
   $$
   2\cos^2 \theta = 1 + \cos 2\theta,
   $$
   with $$\theta = 73^\circ$$, we have
   $$
   2\cos^2 73^\circ = 1 + \cos 146^\circ.
   $$

Thus, the simplified answers are:

- $$ \sin 107^\circ = \sin 73^\circ $$ (approximately $$0.9563$$),
- $$ \cos^{2} 253^\circ + \sin^{2} 557^\circ = 2\cos^2 73^\circ = 1 + \cos 146^\circ $$.
**122.) $$ \sin 107^\circ = \sin 73^\circ $$ (approximately 0.9563)** **128.) $$ \cos^{2} 253^\circ + \sin^{2} 557^\circ = 2\cos^2 73^\circ = 1 + \cos 146^\circ $$**

122.)
128.

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