Question

Given \(\tan \theta=7,\) use trigonometric identities to find the exact value of (a) \(\sec ^{2} \theta\) (b) \(\cot \theta\) (c) \(\cot \left(\frac{\pi}{2}-\theta\right)\) (d) \(\csc ^{2} \theta\)

Short answer

(a) 50, (b) \(\frac{1}{7}\), (c) 7, (d) \(\frac{50}{49}\)

Step-by-step solution

Understanding Given Information

Given that \(\tan \theta = 7\). Use this to find other trigonometric functions using trigonometric identities.

Finding \(\sec^{2} \theta\)

Use the identity \(\sec^{2} \theta = 1 + \tan^{2} \theta\). Substituting \(\tan \theta = 7\), you get \(\sec^{2} \theta = 1 + 7^{2} = 1 + 49 = 50\).

Finding \(\cot \theta\)

Use the identity \(\cot \theta = \frac{1}{\tan \theta}\). Given \(\tan \theta = 7\), you have \(\cot \theta = \frac{1}{7}\).

Finding \(\cot \left(\frac{\pi}{2}-\theta\right)\)

Use the co-function identity: \(\cot \left(\frac{\pi}{2} - \theta\right) = \tan \theta\). Therefore, \(\cot \left(\frac{\pi}{2} - \theta\right) = 7\).

Finding \(\csc^{2} \theta\)

Use the identity \(\csc^{2} \theta = 1 + \cot^{2} \theta\). Substituting \(\cot \theta = \frac{1}{7}\), you get \(\csc^{2} \theta = 1 + \left(\frac{1}{7}\right)^{2} = 1 + \frac{1}{49} = \frac{50}{49}\).

Key concepts

secant squared

To understand \(\text{secant squared}\), we use the trigonometric identity \(\text{sec}^2 \theta = 1 + \tan^2 \theta\). In this exercise, we are given that \(\tan \theta = 7\). By substituting this into the identity, we get:
\[ \text{sec}^2 \theta = 1 + \tan^2 \theta = 1 + 7^2 = 1 + 49 = 50 \]
This shows how secant squared is calculated using the tangent value.
The identity helps simplify our calculations and obtain exact values for trigonometric functions. Always remember, \(\text{sec}^2 \theta\) relates to tangent, making it easier to switch between different trigonometric forms.

cotangent

Next, let's explore the concept of \(\text{cotangent}\). Cotangent is the reciprocal of the tangent function. That is, \(\text{cot} \theta = \frac{1}{\tan \theta}\). Given \(\tan \theta = 7\), we find:
\[ \text{cot} \theta = \frac{1}{7} \]
This simple reciprocal relationship allows us to easily switch between \(\tan\) and \(\text{cot}\).
Understanding this makes working with trigonometric identities more manageable and helps in solving various trigonometric problems.

co-function identities

Co-function identities are useful and powerful tools in trigonometry. They express the relationship between complementary angles. One such identity is for cotangent:
\[ \text{cot} \theta = \tan \theta \]
Specifically, it states:
\[ \text{cot} \theta = \tan\text{(}\left(\frac{\text{\textbackslash pi}}{2}-\theta \right)\left) \text{\textbackslash tan}\theta \]
For this exercise, given \(\tan \theta = 7\), we use the co-function identity:
\[ \text{cot} \theta \text{\textbackslash left}\frac{\text{\textbackslash pi}}{2}-\theta \right)= 7 \] This simplifies the calculation and shows how these identities transform one function into others.

cosecant squared

Finally, let's delve into \(\text{cosecant squared}\). This function connects to cotangent through the identity: \(\text{csc}^2 \theta = 1 + \text{cot}^2 \theta\). Substituting \(\text{cot} \theta = \frac{1}{7}\), we get:
\[ \text{csc}^2 \theta = 1 + \text{\textbackslash left}\frac{1}{7}\right)^2 =1+\text{\textbackslash left}\frac{1}{49}\right)=\frac{50}{49}\]
This identity helps us find the value of \(\text{csc}^2 \theta\) from cotangent. Using these formulas simplifies complex trigonometric expressions. Understanding \(\text{cosecant squared}\) allows for precise and effective solutions, especially when dealing with cotangent values.