Precalculus Trigonometry Syllabus

Trigonometry syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Trigonometry in Precalculus, not a summary of it.

1. 1

   Angles and Their Measure

   3 topics

   • Degree and Radian Measure
   • Arc Length and Sector Area
   • Linear and Angular Speed
2. 2

   Right Triangle Trigonometry

   3 topics

   • The Six Trigonometric Ratios
   • Trigonometric Values of Special Angles
   • Solving Right Triangles and Applications
3. 3

   Trigonometric Functions of Any Angle

   3 topics

   • Unit Circle Definitions
   • Signs in the Four Quadrants
   • Reference Angles and Exact Values
4. 4

   Graphs of Trigonometric Functions

   3 topics

   • Sine and Cosine Graphs
   • Tangent and Cotangent Graphs
   • Secant and Cosecant Graphs
5. 5

   Inverse Trigonometric Functions

   3 topics

   • Inverse Sine, Cosine, and Tangent
   • Restricted Domains and Ranges
   • Compositions with Trig and Inverse Trig

Trigonometry flashcards for Precalculus

20 of 50 cards from the Trigonometry deck — real questions with worked answers.

1. What is the formula to convert an angle from degrees to radians?

   Multiply by $\frac{\pi}{180^{\circ}}$. For example, $\text{radians} = \text{degrees} \times \frac{\pi}{180^{\circ}}$.

2. What is the formula to convert an angle from radians to degrees?

   Multiply by $\frac{180^{\circ}}{\pi}$. For example, $\text{degrees} = \text{radians} \times \frac{180^{\circ}}{\pi}$.

3. How many radians are in one full revolution, and what is the radian measure of $180^{\circ}$ and $90^{\circ}$?

   One full revolution is $2\pi$ radians. $180^{\circ} = \pi$ radians and $90^{\circ} = \frac{\pi}{2}$ radians.

4. Define a radian in terms of arc length and radius.

   One radian is the central angle that subtends an arc equal in length to the radius. In general $\theta = \frac{s}{r}$, where $s$ is arc length and $r$ is radius.

5. What is the formula for arc length $s$ of a circle, and what units must the angle be in?

   $s = r\theta$, where $r$ is the radius and $\theta$ is the central angle measured in radians.

6. What is the formula for the area of a circular sector with central angle $\theta$ (in radians)?

   $A = \frac{1}{2} r^{2} \theta$, where $r$ is the radius and $\theta$ is in radians.

7. Define coterminal angles and how to find one.

   Coterminal angles share the same terminal side. Find them by adding or subtracting full revolutions: $\theta \pm 360^{\circ} k$ (degrees) or $\theta \pm 2\pi k$ (radians) for integer $k$.

8. What is the formula relating linear speed $v$ and angular speed $\omega$?

   $v = r\omega$, where $r$ is the radius and $\omega$ is the angular speed in radians per unit time.

9. How is angular speed $\omega$ defined?

   Angular speed is the rate of change of the central angle: $\omega = \frac{\theta}{t}$, measured in radians per unit time.

10. How is linear speed $v$ defined for a point moving along a circular arc?

    Linear speed is the rate at which arc length is covered: $v = \frac{s}{t}$, where $s$ is arc length and $t$ is time.

11. Define the six trigonometric ratios in a right triangle using opposite, adjacent, and hypotenuse.

    $\sin\theta = \frac{\text{opp}}{\text{hyp}}$, $\cos\theta = \frac{\text{adj}}{\text{hyp}}$, $\tan\theta = \frac{\text{opp}}{\text{adj}}$, $\csc\theta = \frac{\text{hyp}}{\text{opp}}$, $\sec\theta = \frac{\text{hyp}}{\text{adj}}$, $\cot\theta = \frac{\text{adj}}{\text{opp}}$.

12. State the reciprocal identities for the six trigonometric functions.

    $\csc\theta = \frac{1}{\sin\theta}$, $\sec\theta = \frac{1}{\cos\theta}$, $\cot\theta = \frac{1}{\tan\theta}$.

13. Express $\tan\theta$ and $\cot\theta$ as quotients of sine and cosine.

    $\tan\theta = \frac{\sin\theta}{\cos\theta}$ and $\cot\theta = \frac{\cos\theta}{\sin\theta}$.

14. State the Pythagorean identity relating sine and cosine.

    $\sin^{2}\theta + \cos^{2}\theta = 1$.

15. What are the sine, cosine, and tangent of $30^{\circ}$ ($\frac{\pi}{6}$)?

    $\sin 30^{\circ} = \frac{1}{2}$, $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$, $\tan 30^{\circ} = \frac{\sqrt{3}}{3}$.

16. What are the sine, cosine, and tangent of $45^{\circ}$ ($\frac{\pi}{4}$)?

    $\sin 45^{\circ} = \frac{\sqrt{2}}{2}$, $\cos 45^{\circ} = \frac{\sqrt{2}}{2}$, $\tan 45^{\circ} = 1$.

17. What are the sine, cosine, and tangent of $60^{\circ}$ ($\frac{\pi}{3}$)?

    $\sin 60^{\circ} = \frac{\sqrt{3}}{2}$, $\cos 60^{\circ} = \frac{1}{2}$, $\tan 60^{\circ} = \sqrt{3}$.

18. What are the values of $\sin$, $\cos$, and $\tan$ at $0^{\circ}$ and $90^{\circ}$?

    At $0^{\circ}$: $\sin 0 = 0$, $\cos 0 = 1$, $\tan 0 = 0$. At $90^{\circ}$: $\sin 90^{\circ} = 1$, $\cos 90^{\circ} = 0$, $\tan 90^{\circ}$ is undefined.

19. What is a co-function relationship between sine and cosine of complementary angles?

    $\sin\theta = \cos(90^{\circ} - \theta)$ and $\cos\theta = \sin(90^{\circ} - \theta)$; the functions of complementary angles are equal.

20. How do you solve a right triangle given one acute angle and one side?

    Use the trig ratios ($\sin$, $\cos$, $\tan$) to find the remaining sides, subtract the known acute angle from $90^{\circ}$ to get the other acute angle, and use the Pythagorean theorem $a^{2}+b^{2}=c^{2}$ to check or find the last side.

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