1 Solve the equation $$ -0.4 \cos t+1=1.3 $$ where $$ 301^{\circ} \leq t \leq 700^{\circ} $$.

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
-0.4 \cos t + 1 = 1.3 \quad \text{where} \quad 301^{\circ} \leq t \leq 700^{\circ}
$$

**Step 1: Isolate the Cosine Term**

$$
-0.4 \cos t + 1 = 1.3 \
-0.4 \cos t = 1.3 - 1 \
-0.4 \cos t = 0.3 \
\cos t = \frac{0.3}{-0.4} \
\cos t = -0.75
$$

**Step 2: Find the General Solutions**

The general solutions for $$\cos t = -0.75$$ are:

$$
t = 180^{\circ} \pm \arccos(0.75) + 360^{\circ}k \quad \text{for any integer } k
$$

Calculate $$\arccos(0.75)$$:

$$
\arccos(0.75) \approx 41.41^{\circ}
$$

So, the specific solutions are:

$$
t_1 = 180^{\circ} - 41.41^{\circ} = 138.59^{\circ} \
t_2 = 180^{\circ} + 41.41^{\circ} = 221.41^{\circ}
$$

**Step 3: Find Solutions Within the Given Interval $$[301^{\circ}, 700^{\circ}]$$**

Add multiples of $$360^{\circ}$$ to the principal solutions to find values within the interval.

1. For $$t_1 = 138.59^{\circ}$$:
   - Adding $$360^{\circ}$$:
     $$
     138.59^{\circ} + 360^{\circ} = 498.59^{\circ}
     $$
   - Adding another $$360^{\circ}$$ would exceed $$700^{\circ}$$, so only $$498.59^{\circ}$$ is valid.

2. For $$t_2 = 221.41^{\circ}$$:
   - Adding $$360^{\circ}$$:
     $$
     221.41^{\circ} + 360^{\circ} = 581.41^{\circ}
     $$
   - Adding another $$360^{\circ}$$ would exceed $$700^{\circ}$$, so only $$581.41^{\circ}$$ is valid.

**Final Solutions Within $$[301^{\circ}, 700^{\circ}]$$:**

$$
t \approx 498.59^{\circ} \quad \text{and} \quad t \approx 581.41^{\circ}
$$

**Verification:**

To ensure these values satisfy the original equation:

1. For $$t \approx 498.59^{\circ}$$:
   $$
   -0.4 \cos(498.59^{\circ}) + 1 \approx -0.4(-0.75) + 1 = 0.3 + 1 = 1.3
   $$

2. For $$t \approx 581.41^{\circ}$$:
   $$
   -0.4 \cos(581.41^{\circ}) + 1 \approx -0.4(-0.75) + 1 = 0.3 + 1 = 1.3
   $$

Both solutions satisfy the equation.

**Answer:**
The solutions between 301° and 700° are approximately 498.59 ° and 581.41 °.
The solutions between $$301^\circ$$ and $$700^\circ$$ are approximately $$498.59^\circ$$ and $$581.41^\circ$$.

1 Solve the equation where
.

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