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Chapter 5: Problem 6

Fill in the blank to complete the trigonometric identity. \(1+\tan ^{2} u=\)_____

Short Answer

Expert verified
\(\sin(u)\)

Step by step solution

01

Recalling co-function identities

Recall the co-function identity for cosine, which states that \(\cos\left(\frac{\pi}{2}-u\right)=\sin(u)\). Co-function identities can be remembered because sine and cosine are co-functions.
02

Applying co-function identities

Apply the co-function identity to complete the trigonometric identity. So, \(\cos\left(\frac{\pi}{2}-u\right)=\sin(u)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Co-function Identities
Co-function identities are essential tools in trigonometry. These identities reveal relationships between pairs of trigonometric functions, where each pair is known as co-functions.

One of the most common pairs is sine and cosine. Co-function identities are particularly useful when dealing with angles that sum up to \( \left( \frac{\pi}{2} \right) \), or 90 degrees. This is because the sine of an angle is equal to the cosine of its complement and vice versa.

For example:
  • \( \cos\left(\frac{\pi}{2}-u\right)=\sin(u) \)
  • \( \sin\left(\frac{\pi}{2}-u\right)=\cos(u) \)
These identities help simplify trigonometric expressions and solve equations. Knowing these relationships allows us to quickly swap between sine and cosine, making various calculations more manageable.
Sine and Cosine Relationship
Sine and cosine have a deep connection in trigonometry, which can be explored through their co-function identity.

Specifically, these two functions are related by the fact that the cosine of an angle is the same as the sine of its complementary angle. This means that:
  • \( \cos(u) = \sin\left(\frac{\pi}{2}-u\right) \)
  • \( \sin(u) = \cos\left(\frac{\pi}{2}-u\right) \)
These identities help to find unknown angles and simplify expressions.

Visualizing this on a unit circle can further clarify these concepts. The sine function measures the vertical distance from the x-axis, while the cosine measures the horizontal distance. As a result, every point on the unit circle has coordinates that directly relate to these functions.
Angle Subtraction in Trigonometry
Angle subtraction is another crucial tool that plays a significant role in trigonometry. The angle subtraction formulas allow us to break down complex expressions involving trigonometric functions of the sum or difference of angles.

For instance, the formula for cosine subtraction is:
  • \( \cos(a-b) = \cos(a)\cos(b) + \sin(a)\sin(b) \)
In the context of co-function identities, the angle subtraction formula helps us understand why \( \cos\left(\frac{\pi}{2}-u\right) = \sin(u) \).

By recognizing that the expression is a form of angle subtraction, we can effortlessly convert between sine and cosine, easing the process of solving and simplifying equations.

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Most popular questions from this chapter

(a) use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval \([0,2 \pi),\) and (b) solve the trigonometric equation and demonstrate that its solutions are the \(x\) -coordinates of the maximum and minimum points of \(f .\) (Calculus is required to find the trigonometric equation.) Function $$f(x)=\sec x+\tan x-x$$ Trigonometric Equation $$\sec x \tan x+\sec ^{2} x-1=0$$

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the interval \([0,2 \pi)\). $$2 \sin x+\cos x=0$$

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{7 \pi}{12}=\frac{\pi}{3}+\frac{\pi}{4}$$

Write the expression as the sine, cosine, or tangent of an angle. $$\frac{\tan 45^{\circ}-\tan 30^{\circ}}{1+\tan 45^{\circ} \tan 30^{\circ}}$$

Explain in your own words how knowledge of algebra is important when solving trigonometric equations.
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