If tan θ = 15, then what is the value of sec θ?
Solving for sec θ when tan θ = 15 using Trigonometric Identity
The question asks us to find the value of sec θ given that tan θ = 15. To solve this trigonometry problem, we can use a fundamental trigonometric identity that relates tan θ and sec θ.
Understanding the Relationship between tan θ and sec θ
The key trigonometric identity we will use is:
\(\sec^2 \theta = 1 + \tan^2 \theta\)
This identity is derived from the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) by dividing all terms by \(\cos^2 \theta\).
Step-by-Step Calculation
Given that \(\tan \theta = 15\), we can substitute this value into the identity:
\(\sec^2 \theta = 1 + (15)^2\)
Now, we calculate the square of 15:
\(15^2 = 15 \times 15 = 225\)
Substitute this value back into the equation:
\(\sec^2 \theta = 1 + 225\)
\(\sec^2 \theta = 226\)
To find the value of \(\sec \theta\), we take the square root of both sides of the equation:
\(\sec \theta = \pm \sqrt{226}\)
Since the options provided are all positive, we assume that θ is in a quadrant where sec θ is positive (Quadrant I or IV). In standard problems like this without quadrant specification, the positive value is usually expected.
Therefore, taking the positive root, we get:
\(\sec \theta = \sqrt{226}\)
Verification with Options
Let's compare our result with the given options:
- Option 1: \(1/\sqrt{226}\)
- Option 2: \(1/\sqrt{224}\)
- Option 3: \(\sqrt{226}\)
- Option 4: \(\sqrt{224}\)
Our calculated value \(\sqrt{226}\) matches Option 3.
Summary of the Process
We used the relationship between tan θ and sec θ, the identity \(\sec^2 \theta = 1 + \tan^2 \theta\), to find the value of sec θ when tan θ = 15.
| Step | Description | Calculation |
|---|---|---|
| 1 | Start with the identity | \(\sec^2 \theta = 1 + \tan^2 \theta\) |
| 2 | Substitute the given tan θ value | \(\sec^2 \theta = 1 + (15)^2\) |
| 3 | Calculate the square | \(\sec^2 \theta = 1 + 225\) |
| 4 | Simplify | \(\sec^2 \theta = 226\) |
| 5 | Take the square root (positive) | \(\sec \theta = \sqrt{226}\) |
Revision Table: Key Trigonometric Identities
| Identity Name | Formula | Relates |
|---|---|---|
| Pythagorean Identity 1 | \(\sin^2 \theta + \cos^2 \theta = 1\) | sin and cos |
| Pythagorean Identity 2 | \(1 + \tan^2 \theta = \sec^2 \theta\) | tan and sec |
| Pythagorean Identity 3 | \(1 + \cot^2 \theta = \csc^2 \theta\) | cot and csc |
| Reciprocal Identity | \(\sec \theta = 1 / \cos \theta\) | sec and cos |
| Reciprocal Identity | \(\tan \theta = \sin \theta / \cos \theta\) | tan, sin, and cos |
Additional Information: Trigonometric Ratios and Quadrants
Trigonometric ratios like sine, cosine, tangent, secant, cosecant, and cotangent have signs that depend on the quadrant in which the angle \(\theta\) lies.
- In Quadrant I (0° to 90°), all ratios are positive.
- In Quadrant II (90° to 180°), sine and cosecant are positive.
- In Quadrant III (180° to 270°), tangent and cotangent are positive.
- In Quadrant IV (270° to 360°), cosine and secant are positive.
In this problem, since \(\tan \theta = 15\) (a positive value), \(\theta\) could be in Quadrant I or Quadrant III. However, the options only show a positive value for \(\sec \theta\). sec θ is positive in Quadrants I and IV. The only quadrant common to both positive tan θ and positive sec θ is Quadrant I. This supports choosing the positive square root for sec θ.
Important Questions from Trigonometric Functions
If \(\tan \alpha=\frac{1}{7}\), \(\sin \beta=\frac{1}{\sqrt{10}}\); \(0<\alpha, \beta<\frac{\pi}{2}\), then what is the value of cos (α + 2β) ?
What is the period of the function?
What is the value of p + q?
What is the value of pq?
What is pq equal to ?
