If $\cos(x) = -3/5$ and $x$ lies in the third quadrant, what are the values of the other trigonometric functions?

$\sin(x) = -4/5$, $\tan(x) = 4/3$, $\csc(x) = -5/4$, $\sec(x) = -5/3$, $\cot(x) = 3/4$

$\sin(x) = 4/5$, $\tan(x) = -4/3$, $\csc(x) = 5/4$, $\sec(x) = 5/3$, $\cot(x) = -3/4$

$\sin(x) = -4/5$, $\tan(x) = -4/3$, $\csc(x) = -5/4$, $\sec(x) = 5/3$, $\cot(x) = -3/4$

$\sin(x) = 4/5$, $\tan(x) = 4/3$, $\csc(x) = 5/4$, $\sec(x) = -5/3$, $\cot(x) = 3/4$

What is the domain of the function $y = \tan x$?

All real numbers except $x = (2n + 1) \frac{\pi}{2}$, where $n \in \mathbb{Z}$

All real numbers except $x = n\pi$, where $n \in \mathbb{Z}$

All real numbers except $x = (2n + 1)\pi$, where $n \in \mathbb{Z}$

If $y = \tan(x)$, which of the following describes the behavior of the function as $x$ approaches $\frac{\pi}{2}$ from the left?

The function approaches $+\infty$.

The function approaches $-\infty$.

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Trigonometric Functions

Action Plan Notes Questions Mock-Test

Study Action Plan (Sorted by Difficulty)

Graphical Representation of Trigonometric Functions

Accuracy: 44%

Total Attempted41

Total Correct18

Total Incorrect23

Difficulty Level:

High

Periodicity and Reduction of Angles

Accuracy: 50%

Total Attempted32

Total Correct16

Total Incorrect16

Difficulty Level:

Medium

Trigonometric Equations and Solutions

Accuracy: 58%

Learning Objectives

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Learning Objectives

Understand the unit-circle definition of trigonometric functions such as $\sin x$ , $\cos x$ , and $\tan x$ , and their reciprocal functions.
Calculate and convert angle measurements between degrees and radians, and comprehend their implications for trigonometric functions.
Analyze the graphical representation of trigonometric functions, focusing on periodicity, amplitude, and phase shifts.
Determine the domain and range of trigonometric functions, considering restrictions due to asymptotes and periodicity.
Solve trigonometric equations using identities and algebraic manipulation, including finding general solutions.
Derive and apply formulas for trigonometric functions of multiple angles, such as double and triple angle formulas.
Identify the signs of all six trigonometric functions in different quadrants and use this information to deduce other function values.
Explore the periodicity of trigonometric functions and reduce large or negative angles using multiples of $2\pi$ , $\pi$ , or $360^\circ$ .
Apply sum and difference identities for $\sin(x \pm y)$ , $\cos(x \pm y)$ , $\tan(x \pm y)$ , and $\cot(x \pm y)$ in evaluations and proofs.
Utilize product-to-sum and sum-to-product identities in proofs and simplifications of trigonometric expressions.
Prove and simplify expressions using fundamental, compound-angle, multiple-angle, and sum-product identities.

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Revision Notes & Summary

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Chapter Notes

Trigonometric Functions and Identities

Unit-circle definition: For a point $P(a, b)$ on the unit circle, $\cos x = a$ and $\sin x = b$ .
Fundamental identities:
- $\sin^2 x + \cos^2 x = 1$
- $1 + \tan^2 x = \sec^2 x$
- $1 + \cot^2 x = \csc^2 x$
Quadrantal angles:
- $\cos 0^\circ = 1$ , $\sin 0^\circ = 0$
- $\cos \frac{\pi}{2} = 0$ , $\sin \frac{\pi}{2} = 1$
- $\cos \pi = -1$ , $\sin \pi = 0$
- $\cos \frac{3\pi}{2} = 0$ , $\sin \frac{3\pi}{2} = -1$
- $\cos 2\pi = 1$ , $\sin 2\pi = 0$

Angle Measurement and Conversion

Degree measure: One complete revolution is $360^\circ$ .
Radian measure: One complete revolution is $2\pi$ radians.
Conversion:
- $1^\circ = \frac{\pi}{180}$ radians
- $1$ radian $= \frac{180}{\pi}^\circ$

Graphical Representation of Trigonometric Functions

Sine and Cosine:
- Amplitude: 1
- Period: $2\pi$
Tangent and Cotangent:
- Period: $\pi$
Secant and Cosecant:
- Period: $2\pi$

Domain and Range of Trigonometric Functions

Sine and Cosine:
- Domain: All real numbers
- Range: $[-1, 1]$
Tangent and Cotangent:
- Domain: $x \neq (2n+1)\frac{\pi}{2}$ for tangent, $x \neq n\pi$ for cotangent
- Range: All real numbers
Secant and Cosecant:
- Domain: $x \neq (2n+1)\frac{\pi}{2}$ for secant, $x \neq n\pi$ for cosecant
- Range: $(-\infty, -1] \cup [1, \infty)$

Trigonometric Equations and Solutions

General solutions:
- $\sin x = 0 \Rightarrow x = n\pi$
- $\cos x = 0 \Rightarrow x = (2n+1)\frac{\pi}{2}$

Trigonometric Functions of Multiple Angles

Double angle formulas:
- $\sin 2x = 2\sin x \cos x$
- $\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$
- $\tan 2x = \frac{2\tan x}{1 - \tan^2 x}$

Signs and Quadrants of Trigonometric Functions

Quadrant I: All positive
Quadrant II: Sine positive
Quadrant III: Tangent positive
Quadrant IV: Cosine positive

Periodicity and Reduction of Angles

Periodicity:
- $\sin(x + 2n\pi) = \sin x$
- $\cos(x + 2n\pi) = \cos x$
- $\tan(x + n\pi) = \tan x$

Sum and Difference Identities

Sine and Cosine:
- $\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y$
- $\cos(x \pm y) = \cos x \cos y \mp \sin x \sin y$

Product-to-Sum and Sum-to-Product Identities

Identities:
- $\cos x + \cos y = 2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$
- $\sin x + \sin y = 2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$

Trigonometric Identity Proofs and Simplification

Example:
- Prove $\cos(x + y) = \cos x \cos y - \sin x \sin y$ using the unit circle and congruent triangles.

These notes provide a comprehensive overview of the key concepts and formulas related to trigonometric functions, essential for rigorous study and problem-solving in advanced mathematics.

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Practice Test – MCQs, True/False

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Multiple Choice Questions

\frac{24}{7}

\frac{7}{24}

\frac{3}{4}

\frac{4}{3}

Correct Answer: A

Solution:

Using the identity

\tan(2x) = \frac{2 \tan x}{1 - \tan^2 x}

, we have

\tan(2x) = \frac{2 \cdot \frac{3}{4}}{1 - \left(\frac{3}{4}\right)^2} = \frac{\frac{6}{4}}{1 - \frac{9}{16}} = \frac{\frac{6}{4}}{\frac{7}{16}} = \frac{24}{7}

Chapter Concept:

Sum and Difference Identities

\frac{1}{2}

-\frac{1}{2}

\frac{3}{4}

-\frac{3}{4}

Correct Answer: B

Solution:

Using the triple angle formula for cosine,

\cos(3x) = 4\cos^3(x) - 3\cos(x)

. Substituting

\cos(x) = \frac{1}{2}

, we have

\cos(3x) = 4\left(\frac{1}{2}\right)^3 - 3\left(\frac{1}{2}\right) = -\frac{1}{2}

Chapter Concept:

Trigonometric Functions of Multiple Angles

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Trigonometric Functions

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True or False

Correct Answer: True

Solution:

The identity for the cosine of the sum of two angles is

\cos(x + y) = \cos x \cos y - \sin x \sin y

, which is derived using the unit circle and properties of trigonometric functions.

Chapter Concept :

Trigonometric Functions of Multiple Angles

Correct Answer: True

Solution:

The sum and difference identities for trigonometric functions provide formulas for the sine and cosine of sums and differences of angles. Specifically, the identity for sine of a sum is given by

\sin(x + y) = \sin x \cos y + \cos x \sin y

Chapter Concept :

Sum and Difference Identities

Correct Answer: False

Solution:

In the third quadrant, both sine and cosine are negative. Therefore, the sine function is negative.

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Trigonometric Functions

Study Action Plan (Sorted by Difficulty)

Graphical Representation of Trigonometric Functions

High

Periodicity and Reduction of Angles

Medium

Trigonometric Equations and Solutions

Learning Objectives

Learning Objectives

Practice, Analyze & Improve 🚀

Revision Notes & Summary

Chapter Notes

Trigonometric Functions and Identities

Angle Measurement and Conversion

Graphical Representation of Trigonometric Functions

Domain and Range of Trigonometric Functions

Trigonometric Equations and Solutions

Trigonometric Functions of Multiple Angles

Signs and Quadrants of Trigonometric Functions

Periodicity and Reduction of Angles

Sum and Difference Identities

Product-to-Sum and Sum-to-Product Identities

Trigonometric Identity Proofs and Simplification

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Practice Test – MCQs, True/False

Multiple Choice Questions

Q1.If tan⁡x=34\tan x = \frac{3}{4}tanx=43​ and xxx is in the first quadrant, what is tan⁡(2x)\tan(2x)tan(2x)?

Correct Answer: A

Q2.Evaluate cos⁡(3x)\cos(3x)cos(3x) given that cos⁡(x)=12\cos(x) = \frac{1}{2}cos(x)=21​ and xxx is in the first quadrant.

Correct Answer: B

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True or False

Q1.The cosine of the sum of two angles is given by cos⁡(x+y)=cos⁡xcos⁡y−sin⁡xsin⁡y\cos(x + y) = \cos x \cos y - \sin x \sin ycos(x+y)=cosxcosy−sinxsiny.

Correct Answer: True

Q2.The sum and difference identities for sine and cosine can be used to derive the expression sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin(x + y) = \sin x \cos y + \cos x \sin ysin(x+y)=sinxcosy+cosxsiny.

Correct Answer: True

Q3.The sine function is positive in the third quadrant of the unit circle.

Correct Answer: False

Medium

Domain and Range of Trigonometric Functions

Medium

Trigonometric Identity Proofs and Simplification

Medium

Signs and Quadrants of Trigonometric Functions

Medium

Sum and Difference Identities

Medium

Angle Measurement and Conversion

Medium

Trigonometric Functions of Multiple Angles

Low

Trigonometric Functions and Identities

Low

Product-to-Sum and Sum-to-Product Identities

Low

Q3.What is the correct expression for sin⁡(2x)\sin(2x)sin(2x) using double angle formulas?

Correct Answer: A

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Q4.Consider the function y=sin⁡(x)y = \sin(x)y=sin(x). If the function is shifted to the right by π2\frac{\pi}{2}2π​ units, what is the new equation of the function?

Correct Answer: B

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Q5.If cos⁡(x)=−3/5\cos(x) = -3/5cos(x)=−3/5 and xxx lies in the third quadrant, what are the values of the other trigonometric functions?

Correct Answer: A

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Q6.What is the sign of tan⁡x\tan xtanx in the third quadrant?

Correct Answer: A

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Q7.What is the domain of the function y=tan⁡xy = \tan xy=tanx?

Correct Answer: A