Question
2. Use trigonometric identities to simplify the following expressions. (3 pts. each) \[ \text { a. } \sin \alpha \sec \alpha \]
Ask by Dunn Murray. in the United States
Nov 04,2024
Real Tutor Solution
Tutor-Verified Answer
Answer
\( \sin \alpha \sec \alpha = \tan \alpha \)
Solution
Calculate or simplify the expression \( \sin(\alpha) \sec(\alpha) \).
Simplify the expression by following steps:
- step0: Solution:
\(\sin\left(\alpha \right)\sec\left(\alpha \right)\)
- step1: Transform the expression:
\(\sin\left(\alpha \right)\times \frac{1}{\cos\left(\alpha \right)}\)
- step2: Multiply the terms:
\(\frac{\sin\left(\alpha \right)}{\cos\left(\alpha \right)}\)
- step3: Transform the expression:
\(\tan\left(\alpha \right)\)
The trigonometric identity for the expression \( \sin \alpha \sec \alpha \) is \( \tan \alpha \).
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Simplify this solution Extra Insights
To simplify \(\sin \alpha \sec \alpha\), you can use the identity for secant, which states that \(\sec \alpha = \frac{1}{\cos \alpha}\). Therefore, we have: \[ \sin \alpha \sec \alpha = \sin \alpha \cdot \frac{1}{\cos \alpha} = \frac{\sin \alpha}{\cos \alpha} \] This simplifies further to: \[ \tan \alpha \] So, the final answer is \(\tan \alpha\). For a more visual perspective, imagine a right triangle where \(\alpha\) is one of the angles. The opposite side represents \(\sin \alpha\) and the adjacent side relates to \(\cos \alpha\). Thus, understanding trigonometric functions through the lens of triangles makes it easier to grasp their relationships!
