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The Best Complete Trigonometry Guide with Theory & Practice

Learn all the essential Trigonometry theory and formulas, and practice with our extensive question bank. Our practice tool gives you your score, the correct answers, and detailed explanations, helping you maximize your math score. For additional support, please contact us by phone, WhatsApp, or email.
Trigonometry definitions and Formulae
All basic Trigonometry formulae and measure of angles
Free Math Practice Test: Trigonometry Worksheet

Trigonometry Formulae for Sine and Cosine of Supplementary Angles

For any angle \( \theta \) where \( \theta \) is between \( 0^\circ \) and \( 180^\circ \):

  • \(\sin(180^\circ - \theta) = \sin(\theta)\)
  • \(\cos(180^\circ - \theta) = -\cos(\theta)\)


Trigonometry Formulae for Sine and Cosine of Angles Greater than 180°

For any angle \( \theta \) where \( \theta \) is between \( 180^\circ \) and \( 360^\circ \):

  • \(\sin(180^\circ + \theta) = -\sin(\theta)\)
  • \(\cos(180^\circ + \theta) = -\cos(\theta)\)
  • \(\sin(360^\circ - \theta) = -\sin(\theta)\)
  • \(\cos(360^\circ - \theta) = \cos(\theta)\)


Tangent and Cotangent

The tangent and cotangent of angles greater than \( 90^\circ \) can be calculated using:

  • \(\tan(180^\circ - \theta) = -\tan(\theta)\)
  • \(\tan(180^\circ + \theta) = \tan(\theta)\)
  • \(\tan(360^\circ - \theta) = -\tan(\theta)\)
Free Math Practice Test: Trigonometry Worksheet

Cosecant, Secant, and Cotangent

The reciprocal identities extend to angles greater than \( 90^\circ \):

  • \(\csc(180^\circ - \theta) = \csc(\theta)\)
  • \(\sec(180^\circ - \theta) = -\sec(\theta)\)
  • \(\cot(180^\circ - \theta) = -\cot(\theta)\)

For angles greater than \( 180^\circ \):

  • \(\csc(180^\circ + \theta) = -\csc(\theta)\)
  • \(\sec(180^\circ + \theta) = -\sec(\theta)\)
  • \(\cot(180^\circ + \theta) = \cot(\theta)\)


General Angle Formulae

For any angle \( \theta \):

  • \(\sin(360^\circ + \theta) = \sin(\theta)\)
  • \(\cos(360^\circ + \theta) = \cos(\theta)\)
  • \(\tan(360^\circ + \theta) = \tan(\theta)\)
  • \(\csc(360^\circ + \theta) = \csc(\theta)\)
  • \(\sec(360^\circ + \theta) = \sec(\theta)\)
  • \(\cot(360^\circ + \theta) = \cot(\theta)\)


Negative Angle Identities

For any angle \( \theta \):

  • \(\sin(-\theta) = -\sin(\theta)\)
  • \(\cos(-\theta) = \cos(\theta)\)
  • \(\tan(-\theta) = -\tan(\theta)\)
  • \(\csc(-\theta) = -\csc(\theta)\)
  • \(\sec(-\theta) = \sec(\theta)\)
  • \(\cot(-\theta) = -\cot(\theta)\)

Trigonometry Practice Questions: Easy Level

Trigonometry Practice Questions: Easy Level

Q1. Find the value of \( \sin 30^\circ \):

Refer the Trigonometry Formulae: \( \sin 30^\circ = \frac{1}{2} = 0.5 \)

Q2. Find the value of \( \cos 30^\circ \):

Refer the Trigonometry Formulae: \( \cos 0^\circ = 1 \)

Q3. Find the value of \( \tan 45^\circ \):

Refer the Trigonometry Formulae: \( \tan 45^\circ = 1 \)

Q4. Find the value of \( \sin 90^\circ \):

Refer the Trigonometry Formulae: \( \sin 90^\circ = 1 \)

Q5. Find the value of \( \cos 90^\circ \):

Refer the Trigonometry Formulae: \( \cos 90^\circ = 0 \)

Q6. Find the value of \( \tan 0^\circ \):

Refer the Trigonometry Formulae: \( \tan 0^\circ = 0 \)

Q7. If \( \sin \theta = \frac{1}{2} \), find \( \theta \) (0° ≤ θ ≤ 360°):

Refer the Trigonometry Formulae: \( \sin 30^\circ = \frac{1}{2} = 0.5; Hence, \theta = 30^\circ \)

Q8. If \( \cos \theta = \frac{1}{2} \), find \( \theta \) (0° ≤ θ ≤ 360°)

Refer the Trigonometry Formulae: \( \cos 60^\circ = \frac{1}{2} = 0.5; Hence, \theta = 60^\circ \)

Q9. If \( \tan \theta = 1 \), find \( \theta \) (0° ≤ θ ≤ 360°):

Refer the Trigonometry Formulae: \( \tan 45^\circ = 1; Hence, \theta = 45^\circ \)

Q10. Find the value of \( \sin 60^\circ \):

Refer the Trigonometry Formulae: \( \sin 60^\circ = \frac{\sqrt(3)}{2} \). In the box, the right answer is sqrt3/2

Trigonometry Practice Questions: Medium Level

Trigonometry Practice Questions: Medium Level

Q1. Find the value of \( \sin 120^\circ \):

\( \sin \theta = sin (180-\theta), \sin 120^\circ = \sin(180-120) = sin 60^\circ = \frac{\sqrt(3)}{2} \); P.S.: In the box, the right answer is sqrt3/2

Q2. Find the value of \( \cos 240^\circ \):

\( \cos(180+\theta) = -cos(\theta). Therefore, \cos 240^\circ = \cos(180+120) = -cos 60^\circ = \frac{\sqrt(3)}{2} \). P.S.: In the box, the right answer is sqrt3/2

Q3. Find the value of \( \tan 135^\circ \):

\( \tan(180-\theta) = -tan(\theta). Therefore, \cos 135^\circ = \tan(180-135) = -tan 45^\circ = -1 \)

Q4. Solve \( \sin 60^\circ - \cos 30^\circ + 7\tan 45^\circ \) :

\( \sin 60^\circ = \cos 30^\circ = \frac{\sqrt(3)}{2}; \tan 45^\circ = 1; \sin 60^\circ - \cos 30^\circ + 7\tan 45^\circ = \frac{\sqrt(3)}{2} - \frac{\sqrt(3)}{2} +7(1) = 7 \)

Q5. Solve \( \frac{\cos 35^\circ}{ \sin 55^\circ} \)

Using the formula: \( sin\theta=cos(90-\theta). Hence, \sin 55^\circ=\cos 35^\circ. Therefore, \frac{\cos 35^\circ}{ \sin 55^\circ} = \frac{\cos 35^\circ}{ cos 35^\circ} =1 \)

Q6. If \( \sin\theta = \frac{3}{5}, find \cos\theta \):

Using the formula: \( \sin^2\theta + \cos^2\theta = 1; \) Substituting the value for \( \sin\theta = \frac{3}{5}, we get \cos\theta = \frac{4}{5} \):

Q7. If \( \sin \theta = \frac{1}{2} \), find \( \theta \) (0° ≤ θ ≤ 360°):

Refer the Trigonometry Formulae: \( \sin 30^\circ = \frac{1}{2} = 0.5; Hence, \theta = 30^\circ \)

Q8. Solve \( 5\tan45^\circ + \sin 45^\circ - \cos 45^\circ + 7\tan 45^\circ \) :

\( \sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt2}; \tan 45^\circ = 1; \) Therefore \( 5\tan45^\circ + \sin 45^\circ - \cos 45^\circ + 7\tan 45^\circ = 5(1)+ \frac{1}{\sqrt2} - \frac{1}{\sqrt2} +7(1) = 12 \)

Q9. If \( \sin\theta = \frac{24}{25} \), find \( \tan\theta \).

Use the formula \( \sin^2\theta + \cos^2\theta = 1; \) Substituting the value for \( \sin\theta = \frac{24}{25}, we get \cos\theta = \frac{7}{25} and \tan\theta=\frac{\sin\theta}{\cos\theta} = \frac{\frac{24}{25}}{\frac{25}{7}} = \frac{24}{7} \):

Q10. Solve \( 10\sin 30^\circ + 10\cos 60^\circ + 10\tan 45^\circ \) :

\( \sin 30^\circ = \cos 60^\circ = \frac{1}{2}; \tan 45^\circ = 1; Therefore: 10\sin 30^\circ + 10\cos 60^\circ + 10\tan 45^\circ = 10*\frac{1}{2} + 10*\frac{1}{2} + 10(1) = 5+5+10 = 20 \)

Trigonometry Practice Questions: Hard Level

Free Math Practice Test: Trigonometry Worksheet

1. Prove that \( \sin(90^\circ - \theta) = \cos \theta \):


2. Prove that \( \cos(90^\circ - \theta) = \sin \theta \):


3. Simplify \( \tan(\theta + 45^\circ) \):


4. Prove that \( \sin 2\theta = 2 \sin \theta \cos \theta \):


5. Prove that \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \):


6. Simplify \( \tan 2\theta \):


7. Prove that \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \):


8. Prove that \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \):


9. Simplify \( \cot(90^\circ - \theta) \):


10. Prove that cosec\( (90^\circ - \theta) = \sec \theta \):

Why is Trigonometry important?

  1. Foundation for Advanced Math: Essential for calculus, linear algebra, and differential equations.

  2. Real-World Applications: Used in engineering, physics, architecture, and everyday measurements.

  3. Problem-Solving Skills: Enhances logical reasoning and critical thinking.

  4. Scientific Understanding: Crucial for studying waves, oscillations, and physical phenomena.

  5. Technological Relevance: Key for careers in STEM fields.

 

Why is Trigonometry important for exams?

  1. Comprehensive Assessment: Evaluates overall mathematical understanding and application skills.

  2. Curriculum Requirement: Ensures mastery of essential topics before higher education.

  3. Skill Measurement: Tests algebraic manipulation, spatial reasoning, and analytical thinking.

  4. Standardized Test Prep: Prepares students for SAT, ACT, and other entrance exams.

  5. Holistic Development: Promotes mathematical reasoning and complex problem-solving.

 

Conclusion for Trigonometry:

Trigonometry is vital for foundational knowledge, practical applications, and skill development, making it crucial in both education and exams.

Improved Guide to Solving Trigonometry Problems

Solving trigonometry questions typically involves the following steps:

  1. Understand the problem: Read the question and identify givens, unknowns and what needs to be found.

  2. Draw a diagram: If the questions deals with geometry, draw a diagram for better understanding. Label angles, sides, and information. This will provide a holistic view.

  3. Choose trigonometric functions: Use the trigonometric functions to convert all the information into sine, cosine. Determine which functions can be applied based on the given information and what needs to be found.

  4. Create an equation: Relate known and unknown values. Ensure the equation accurately reflects the problem’s conditions.

  5. Solve for the unknown: Use algebra or inverse trig functions.

  6. Check the solution: Verify your solution by plugging your answers back into the original equation or use the solution it in the context of the problem.

  7. Apply identities (if needed): Simplify expressions or find alternative solutions.

  8. Use a calculator: Find values for non-exact answers.

  9. Practice: The more you practice, the better you will have a grasp on solving Trigonometric problems

 

Trigonometry might seem tricky at first, but you can definitely master it!

Just like other math topics, practice is key. Even though you’ll learn new words and symbols, the thinking process is similar to what you already know from algebra and geometry.

The secret to success? Break down problems into smaller steps and try to picture what’s happening.

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