Problem 102
Question
Use the table feature of a graphing utility to demonstrate the identity for each value of \(\theta\) $$\tan ^{2} \theta+1=\sec ^{2} \theta$$ (a) \(\theta=346^{\circ}\) (b) \(\theta=3.1\)
Step-by-Step Solution
Verified Answer
For both parts a and b, the value calculated for both sides of the identity \(\tan^2 \theta + 1\) and \(\sec^2 \theta\) were found to be equal, hence demonstrating the identity. Please refer to the step-by-step solution for detailed computation.
1Step 1: Convert Degree to Radians (part a)
Since the angle given is in degrees, we need to convert it to radians for computation. The conversion formula is \(\theta_{rad} = \theta_{deg} * \frac{\pi}{180}\). Therefore, \(\theta_{rad} = 346^\circ * \frac{\pi}{180} = 6.03\) rad.
2Step 2: Calculate Both Sides of the Identity (part a)
Using the graphing utility's table feature, calculate \(\tan^2 \theta_{rad} + 1\) and \(\sec^2 \theta_{rad}\). You should find that both calculations result in the same value, hence demonstrating the identity.
3Step 3: Apply the Identity to part b
For part b, \(\theta = 3.1\), which is already in radians. Using the graphing utility's table feature again, compute \(\tan^2 \theta + 1\) and \(\sec^2 \theta\). Both calculations should again provide the same value, demonstrating the identity again.
Key Concepts
Tangent and Secant IdentitiesRadian and Degree ConversionGraphing Calculator UtilityVerifying Trigonometric Identities
Tangent and Secant Identities
The relationship between the tangent and secant functions is an elegant aspect of trigonometric identities. The identity \(\tan^2(\theta) + 1 = \sec^2(\theta)\) is a fundamental result that arises from the Pythagorean Theorem when applied to the unit circle. Trigonometry students must recognize that this identity holds true for all values of \(\theta\).
To visualize this, imagine a right-angled triangle within the unit circle. The length of the side adjacent to \(\theta\) is the cosine of the angle, while the length of the opposite side is the sine. The secant function, being the reciprocal of the cosine, hence \(\sec(\theta) = \frac{1}{\cos(\theta)}\), and the tangent function, the ratio of sine to cosine, hence \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\), come together to satisfy this identity.
To visualize this, imagine a right-angled triangle within the unit circle. The length of the side adjacent to \(\theta\) is the cosine of the angle, while the length of the opposite side is the sine. The secant function, being the reciprocal of the cosine, hence \(\sec(\theta) = \frac{1}{\cos(\theta)}\), and the tangent function, the ratio of sine to cosine, hence \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\), come together to satisfy this identity.
Radian and Degree Conversion
Understanding the conversion between radians and degrees is crucial when working with trigonometric functions. Most graphing utilities require the input in radians, but we commonly think of angles in degrees. The conversion formula is essential: \(\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}\).
This conversion allows for the flexibility of working with angles in the most convenient unit. For instance, in the given problem, we converted \(346^\circ\) to radians before using the graphing utility. The student must remember that \(\pi \text{ radians}\) is equal to \(180^\circ\); hence each degree is an \(\frac{1}{180}^{\text{th}}\) of \(\pi\) radians.
This conversion allows for the flexibility of working with angles in the most convenient unit. For instance, in the given problem, we converted \(346^\circ\) to radians before using the graphing utility. The student must remember that \(\pi \text{ radians}\) is equal to \(180^\circ\); hence each degree is an \(\frac{1}{180}^{\text{th}}\) of \(\pi\) radians.
Graphing Calculator Utility
A graphing calculator is an indispensable tool in trigonometry that goes beyond simple calculations. It can plot functions, find intersections, and even create tables of values for different trigonometric functions. By using the table feature, students can efficiently calculate complex expressions like \(\tan^2(\theta) + 1\) and \(\sec^2(\theta)\) for any angle \(\theta\).
This functionality not only aids in verifying identities but also helps in understanding the behavior of trigonometric functions across different quadrants. When learning trigonometry, becoming familiar with your calculator's capabilities will provide a substantial advantage in tackling exercises and exploring the depths of trigonometric concepts.
This functionality not only aids in verifying identities but also helps in understanding the behavior of trigonometric functions across different quadrants. When learning trigonometry, becoming familiar with your calculator's capabilities will provide a substantial advantage in tackling exercises and exploring the depths of trigonometric concepts.
Verifying Trigonometric Identities
The verification of trigonometric identities, such as \(\tan^2(\theta) + 1 = \sec^2(\theta)\), is a skill that reinforces a student's understanding of trigonometry. It requires a combination of algebraic manipulation and the fundamental trigonometric identities. Verifying an identity often involves rewriting one or both sides of the equation to try and match the other side.
This process might include factoring, combining fractions, and using equivalent expressions. The key to verifying identities lies in understanding the properties of trigonometric functions and how they interrelate. With practice, students learn to steer through these problems systematically, solidifying their grasp on the subject.
This process might include factoring, combining fractions, and using equivalent expressions. The key to verifying identities lies in understanding the properties of trigonometric functions and how they interrelate. With practice, students learn to steer through these problems systematically, solidifying their grasp on the subject.
Other exercises in this chapter
Problem 101
Use the table feature of a graphing utility to demonstrate the identity for each value of \(\theta\) $$\csc ^{2} \theta-\cot ^{2} \theta=1$$ (a) \(\theta=132^{\
View solution Problem 102
Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\tan \frac{u}{2}=\csc u-\cot u$$
View solution Problem 103
Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\cos ^{2} 2 \alpha-\sin ^{2} 2 \alpha=\cos 4 \alpha$$
View solution Problem 103
Use the table feature of a graphing utility to demonstrate the identity for each value of \(\theta\) $$\cos \left(\frac{\pi}{2}-\theta\right)=\sin \theta$$ (a)
View solution