Question

If \(\sin \theta=0.3,\) find the exact value of \(\sin \theta+\cos \left(\frac{\pi}{2}-\theta\right)\) Find an acute angle \(\theta\) that satisfies the equation

Short answer

\( 0.6 \)

Step-by-step solution

Identify the Given Information

Given that \( \sin \theta = 0.3 \). We need to find the value of \( \sin \theta + \cos \left( \frac{\pi}{2} - \theta \right) \).

Recall the Co-Function Identity

Using the co-function identity for cosine and sine, recall that \( \cos \left( \frac{\pi}{2} - \theta \right) = \sin \theta \).

Substitute the Given Values

Substitute the given values into the expression: \( \sin \theta + \cos \left( \frac{\pi}{2} - \theta \right) = \sin \theta + \sin \theta \). Since \( \sin \theta = 0.3 \), we get \( 0.3 + 0.3 \).

Simplify the Expression

Simplify the expression \( 0.3 + 0.3 = 0.6 \).

Verify the Acute Angle Condition

Since \( \sin \theta = 0.3 \), \( \theta \) must be an acute angle in the interval \( 0 < \theta < \frac{\pi}{2} \). Given \( \sin \theta = 0.3 \, \theta \) satisfies the acute angle condition.

Key concepts

Sine Function

The sine function is a fundamental trigonometric function. It relates the angle in a right triangle to the ratio of the opposite side over the hypotenuse. For any acute angle \( \theta \), the sine value can range from 0 to 1.
In our exercise, we have \( \sin \theta = 0.3 \). This means for our specific angle \( \theta \), the ratio of the opposite side to the hypotenuse is 0.3.
The sine function is crucial in many areas of mathematics and physics, providing a way to describe oscillatory behavior.

Co-Function Identities

Co-function identities are relationships between trigonometric functions of complementary angles. Complementary angles add up to \( \frac{\pi}{2} \) (or 90 degrees).
One such co-function identity is \( \cos \left( \frac{\pi}{2} - \theta \right) = \sin \theta \). This identity indicates that the cosine of the complement of an angle equals the sine of the angle itself.
In this exercise, we use this identity to show that \( \cos \left( \frac{\pi}{2} - \theta \right) = \sin \theta \). By recognizing this relationship, we easily substitute and simplify our expression to find the required value.

Acute Angle

An acute angle is any angle that is between 0 and 90 degrees (or 0 and \( \frac{\pi}{2} \) radians). Acute angles are important in trigonometry as the values of trigonometric functions are well-defined and always positive within this range.
In our solution, we verify that \( \theta \) is an acute angle because one of the properties of sine is that it returns a positive value less than 1 when the angle is acute. Given \( \sin \theta = 0.3 \), we confirm that \( \theta \) lies in the interval \( 0 < \theta < \frac{\pi}{2} \), ensuring it is acute.
This step helps us validate the context and correctness of the solution.