Question

If cos theta = √2/2, what are the values of sin theta and tan theta?

* sin theta = √2/2 ; tan theta = -1

* sin theta = -√2/2 ; tan theta = 1

* sin theta = √2/2 ; tan theta = -√2

* sin theta = -√2/2 ; tan theta = -1

Answer 1

For cos theta = √2/2, the values of sin theta and tan theta in the first quadrant are also √2/2 and 1, respectively. This is obtained by using the Pythagorean identity and definition of tangent.

Step-by-step explanation:

If cos theta = √2/2, we know that this value corresponds to an angle of 45° (or π/4 radians) in the unit circle where both the sine and cosine values are positive and equal. To find the values of sin theta and tan theta here's what we need to consider:

• The Pythagorean identity: sin² theta + cos² theta = 1
• The definition of tangent: tan theta = sin theta / cos theta

Using the Pythagorean identity:

1. sin² theta = 1 - cos² theta
2. sin² theta = 1 - (√2/2)²
3. sin² theta = 1 - 1/2
4. sin² theta = 1/2
5. sin theta = ±√(1/2)

Since we are dealing with the first quadrant where both sine and cosine are positive:

1. sin theta = √(1/2) = √2/2

Now for tangent:

1. tan theta = sin theta / cos theta
2. tan theta = (√2/2) / (√2/2)
3. tan theta = 1

Therefore, the values are:

• sin theta = √2/2
• tan theta = 1