As the angle increases beyond 45 degrees, what happens to the tangent value?
Tangent value increases above 1
To understand what happens to the tangent value as the angle increases beyond 45 degrees, let's investigate the properties of the tangent function in trigonometry:
- The tangent function, denoted as \(\tan(\theta)\), is defined as the ratio of the sine and cosine functions: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).
- At \(45^\circ\), the value of tangent is exactly 1. This is because both sine and cosine of \(45^\circ\) are equal, leading to: \(\tan(45^\circ) = \frac{1}{1} = 1\).
- For angles greater than \(45^\circ\), the sine value becomes larger than the cosine value because \(\sin(\theta)\) increases and \(\cos(\theta)\) decreases as the angle increases towards 90 degrees.
- Therefore, when \(\theta > 45^\circ\), the ratio \(\frac{\sin(\theta)}{\cos(\theta)}\) (tangent value) becomes greater than 1.
Thus, as the angle increases beyond \(45^\circ\), the tangent value increases and becomes greater than 1. Therefore, the correct option is:
Tangent value increases above 1
Other options are incorrect because:
The tangent value does not become zero or remain constant. Instead, it continues to increase as the angle approaches 90 degrees.
The tangent value does not decrease below 1 for angles greater than 45 degrees.
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