Domanda del libro di testo
Concept Check Work each problem. Without using a calculator, determine which of the following numbers is closest to sin 115°: -0.9, -0.1, 0, 0.1, or 0.9.
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Ch. 2 - Acute Angles and Right Triangles
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891
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TOC.tableOfContents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 87
Concept Check Work each problem. For what angles θ between 0° and 360° is cos θ = sin θ true?
Guida verificata passo dopo passo1
Recall the fundamental trigonometric identity that relates sine and cosine: \(\sin \theta = \cos \theta\).
Rewrite the equation \(\cos \theta = \sin \theta\) by dividing both sides by \(\cos \theta\), assuming \(\cos \theta \neq 0\), to get \(1 = \tan \theta\).
Express the equation in terms of tangent: \(\tan \theta = 1\).
Find the general solutions for \(\theta\) where \(\tan \theta = 1\) within the interval \(0^\circ \leq \theta < 360^\circ\). Recall that \(\tan \theta = 1\) at angles where \(\theta = 45^\circ + k \times 180^\circ\), where \(k\) is an integer.
Identify the specific angles between \(0^\circ\) and \(360^\circ\) by substituting values of \(k\) to find all solutions in the given range.
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Relationship Between Sine and Cosine Functions
Sine and cosine are fundamental trigonometric functions representing ratios of sides in a right triangle or coordinates on the unit circle. Understanding how their values compare at different angles is key to solving equations like cos θ = sin θ.
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5:33
Period of Sine and Cosine Functions
Unit Circle and Angle Measurement
The unit circle is a circle with radius 1 centered at the origin, where angles correspond to points with coordinates (cos θ, sin θ). Knowing how to interpret angles between 0° and 360° on the unit circle helps identify where sine and cosine values are equal.
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06:11Introduction to the Unit Circle
Solving Trigonometric Equations
Solving equations like cos θ = sin θ involves algebraic manipulation and using identities or geometric interpretations. Recognizing that cos θ = sin θ implies tan θ = 1 allows finding specific angle solutions within the given interval.
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4:34How to Solve Linear Trigonometric Equations