Textbook Question
Find the remaining five trigonometric functions of θ.
csc θ = -5/2, θ in quadrant III
45
views
6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
3:35 minutesProblem 1.2.67
Textbook Question
In Exercises 63–68, find the exact value of each expression. Do not use a calculator. csc 37° sec 53° - tan 53° cot 37°
Verified step by step guidance1
Recall the complementary angle relationships: since 37° and 53° add up to 90°, we have \( \sin 37^\circ = \cos 53^\circ \) and \( \cos 37^\circ = \sin 53^\circ \). This will help simplify the trigonometric expressions.
Rewrite each trigonometric function in terms of sine and cosine: \( \csc 37^\circ = \frac{1}{\sin 37^\circ} \), \( \sec 53^\circ = \frac{1}{\cos 53^\circ} \), \( \tan 53^\circ = \frac{\sin 53^\circ}{\cos 53^\circ} \), and \( \cot 37^\circ = \frac{\cos 37^\circ}{\sin 37^\circ} \).
Substitute these into the expression: \( \csc 37^\circ \sec 53^\circ - \tan 53^\circ \cot 37^\circ = \frac{1}{\sin 37^\circ} \times \frac{1}{\cos 53^\circ} - \frac{\sin 53^\circ}{\cos 53^\circ} \times \frac{\cos 37^\circ}{\sin 37^\circ} \).
Use the complementary angle identities to replace \( \cos 53^\circ \) with \( \sin 37^\circ \) and \( \sin 53^\circ \) with \( \cos 37^\circ \), simplifying the expression to terms involving only \( \sin 37^\circ \) and \( \cos 37^\circ \).
Simplify the resulting expression by combining fractions and canceling common terms to find the exact value without using a calculator.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Reciprocal Trigonometric Functions
Cosecant (csc) and secant (sec) are the reciprocals of sine and cosine, respectively. Specifically, csc θ = 1/sin θ and sec θ = 1/cos θ. Understanding these relationships helps simplify expressions involving these functions without a calculator.
Recommended video:
6:04
Introduction to Trigonometric Functions
Complementary Angle Relationships
Angles like 37° and 53° are complementary because they add up to 90°. Trigonometric functions of complementary angles are related, for example, sin(37°) = cos(53°) and tan(37°) = cot(53°). This property is useful for rewriting and simplifying expressions.
Recommended video:
3:35Intro to Complementary & Supplementary Angles
Simplification of Trigonometric Expressions
Simplifying expressions involves substituting reciprocal and complementary angle identities and combining terms. Recognizing patterns and using fundamental identities allows finding exact values without a calculator, which is essential for solving the given problem.
Recommended video:
6:36Simplifying Trig Expressions