if 0<t< π/2 and cos(t)= 4/5: evaluate: sin(t+ 20π) + sin(t-6π) + tan(t+5π)

We are given that \( 0 < t < \frac{\pi}{2} \) and \( \cos(t) = \frac{4}{5} \). Our goal is to evaluate the expression:

\[
\sin(t + 20\pi) + \sin(t - 6\pi) + \tan(t + 5\pi)
\]

### Step 1: Find \( \sin(t) \) and \( \tan(t) \)

Given \( \cos(t) = \frac{4}{5} \), we can find \( \sin(t) \) using the Pythagorean identity:

\[
\sin^2(t) + \cos^2(t) = 1
\]

Substituting \( \cos(t) = \frac{4}{5} \):

\[
\sin^2(t) + \left(\frac{4}{5}\right)^2 = 1
\]
\[
\sin^2(t) + \frac{16}{25} = 1
\]
\[
\sin^2(t) = 1 - \frac{16}{25} = \frac{9}{25}
\]
\[
\sin(t) = \frac{3}{5}
\]

Since \( 0 < t < \frac{\pi}{2} \), \( \sin(t) \) is positive, so \( \sin(t) = \frac{3}{5} \).

Now, find \( \tan(t) \):

\[
\tan(t) = \frac{\sin(t)}{\cos(t)} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4}
\]

### Step 2: Simplify the trigonometric expressions

- **For \( \sin(t + 20\pi) \):** Since sine has a period of \( 2\pi \), we know:
  \[
  \sin(t + 20\pi) = \sin(t)
  \]
  Thus, \( \sin(t + 20\pi) = \frac{3}{5} \).

- **For \( \sin(t - 6\pi) \):** Similarly, since \( \sin(t) \) repeats every \( 2\pi \), we have:
  \[
  \sin(t - 6\pi) = \sin(t)
  \]
  Thus, \( \sin(t - 6\pi) = \frac{3}{5} \).

- **For \( \tan(t + 5\pi) \):** Tangent has a period of \( \pi \), so:
  \[
  \tan(t + 5\pi) = \tan(t)
  \]
  Thus, \( \tan(t + 5\pi) = \frac{3}{4} \).

### Step 3: Evaluate the expression

Now, we can sum the values:

\[
\sin(t + 20\pi) + \sin(t - 6\pi) + \tan(t + 5\pi)
\]
\[
= \frac{3}{5} + \frac{3}{5} + \frac{3}{4}
\]

### Step 4: Compute the final value

First, add the fractions \( \frac{3}{5} + \frac{3}{5} = \frac{6}{5} \). Now add \( \frac{6}{5} + \frac{3}{4} \). The least common denominator (LCD) of 5 and 4 is 20, so:

\[
\frac{6}{5} = \frac{24}{20}, \quad \frac{3}{4} = \frac{15}{20}
\]
\[
\frac{24}{20} + \frac{15}{20} = \frac{39}{20}
\]

Thus, the final value is:

\[
\boxed{\frac{39}{20}}
\]

Would you like any additional details or clarifications?

Here are five related questions for further practice:
1. What is the period of the cosine function, and how does it apply here?
2. How would the result change if \( t \) were in the second quadrant?
3. What is the value of \( \cos(t + 5\pi) \) in this scenario?
4. How does changing the argument of sine and tangent functions by multiples of \( \pi \) affect the result?
5. Can you derive the value of \( \sin(2t) \) from this information?

**Tip:** Remember that trigonometric functions are periodic, which means shifting the argument by multiples of \( 2\pi \) (for sine and cosine) or \( \pi \) (for tangent) leaves the function unchanged.

If 0 < t < π/2 and cos(t) = 4/5, evaluate: sin(t + 20π) + sin(t - 6π) + tan(t + 5π)

Evaluate the trigonometric expression sin(t + 20π) + sin(t - 6π) + tan(t + 5π), where cos(t) = 4/5 and 0 < t < π/2. Learn how to apply the Pythagorean identity and periodic properties of trigonometric functions. Suitable for students in Grades 10-12.

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