Questions: If sin θ=0.4, find the exact value of sin θ+cos (π/2-θ). sin θ+cos (π/2-θ)= (Type an integer or a decimal.)

If sin θ=0.4, find the exact value of sin θ+cos (π/2-θ). sin θ+cos (π/2-θ)= (Type an integer or a decimal.)
Transcript text: If $\sin \theta=0.4$, find the exact value of $\sin \theta+\cos \left(\frac{\pi}{2}-\theta\right)$. \[ \sin \theta+\cos \left(\frac{\pi}{2}-\theta\right)= \] $\square$ (Type an integer or a decimal.)
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Solution

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Solution Steps

To solve the given problem, we need to use the trigonometric identity that relates sine and cosine functions. Specifically, we use the identity \(\cos \left(\frac{\pi}{2} - \theta\right) = \sin \theta\). This allows us to simplify the expression \(\sin \theta + \cos \left(\frac{\pi}{2} - \theta\right)\).

Solution Approach
  1. Recognize that \(\cos \left(\frac{\pi}{2} - \theta\right) = \sin \theta\).
  2. Substitute \(\cos \left(\frac{\pi}{2} - \theta\right)\) with \(\sin \theta\).
  3. Simplify the expression to find the exact value.
Step 1: Given Information

We are given that \( \sin \theta = 0.4 \).

Step 2: Use Trigonometric Identity

We apply the identity \( \cos \left(\frac{\pi}{2} - \theta\right) = \sin \theta \). Therefore, we can rewrite the expression:

\[ \sin \theta + \cos \left(\frac{\pi}{2} - \theta\right) = \sin \theta + \sin \theta = 2 \sin \theta \]

Step 3: Calculate the Value

Substituting the given value of \( \sin \theta \):

\[ 2 \sin \theta = 2 \times 0.4 = 0.8 \]

Final Answer

Thus, the exact value of \( \sin \theta + \cos \left(\frac{\pi}{2} - \theta\right) \) is

\[ \boxed{0.8} \]

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