44 Questions
What is the amplitude of a periodic function?
The height from the centre to the maximum or minimum values
Sin is positive in the 2nd quadrant according to the ASTC rule.
True
What is the angle that gives θ in the triangle in the 3rd quadrant?
180° + θ
ASTC rule stands for 'All Stations To ________'.
Central
Match the trigonometric functions with their inverse functions:
sin = sin⁻¹ cos = cos⁻¹ tan = tan⁻¹
What is the value of $a$ in the Pythagorean theorem equation $a^2 + 4 = 49$?
5
Which trigonometric ratio is equivalent to $\cot \theta$?
$\tan \theta$
For any value of $\theta$, $\cos^2 \theta + \sin^2 \theta = ___$
1
Match the following Pythagorean identities:
$\cos^2 \theta + \sin^2 \theta$ = 1 $1 + \tan^2 \theta$ = $\sec^2 \theta$ $1 + \cot^2 \theta$ = $\csc^2 \theta$
What is the value of $m$ in the equation $\sec 55^\circ = \csc (2m - 15)^\circ$?
25
Find all quadrants where cos θ > 0, tan θ > 0, and sin θ > 0. (Select all that apply)
1st quadrant
Which quadrant is the angle 240° in?
3rd quadrant
Find the exact value of cos 240°.
-1/2
Which quadrant is the angle 315° in?
4th quadrant
Find the exact value of sin 315°.
1/√2
Which quadrant is the angle 120° in?
2nd quadrant
Find the exact value of tan 120°.
√3
Which quadrant is the angle -225° in?
3rd quadrant
Find the exact value of sin (-225°).
-1/√2
Which quadrant is the angle -330° in?
4th quadrant
Find the exact value of cos (-330°).
-√3/2
Find the exact value of tan (-120°).
√3
Write cos 2x sin 5x as a sum of trigonometric ratios.
sin 7x + sin 3x
Find the exact value of sin 75° sin 15°.
1/4
Find the exact value of 2tan15° / (1 + tan²15°).
1
Prove that cot(A/2) - 2cotA = tan(A/2).
tan(A/2)
Find the exact value of sin 30°.
1/2
Find an expression for sin (x + y) + sin (x − y).
2sin(x)cos(y)
Write an expression for cos (x + y) − cos (x − y).
2sin(x)sin(y)
In which quadrant is the angle 3π/4 located?
2nd quadrant
Find the exact value of cos(3π/4).
-√2/2
Simplify: 2cos(3x)sin(3x).
sin(6x)
Show that $\frac{5\pi}{6} = \frac{\pi}{6}$.
$5\pi = \pi + 6$
In which quadrant is the angle $\frac{5\pi}{6}$?
2nd quadrant
Find the exact value of $\sin \frac{5\pi}{6}$.
-$\frac{1}{2}$
Show that $\frac{7\pi}{4} = 2\pi - \frac{\pi}{4}$.
$7\pi = 8\pi - 4$
In which quadrant is the angle $\frac{7\pi}{4}$?
1st quadrant
Find the exact value of $\tan \frac{7\pi}{4}$.
1
What is the domain of the function y = tan(x)?
(-∞, ∞) except for odd multiples of 90° (90°, 270°, 450°, ...)
What is the range of the function y = tan(x)?
(-∞, ∞)
What is the domain of the function y = cosec(x)?
(-∞, ∞) except for 0°, 180°, 360°, ...
What is the range of the function y = cosec(x)?
(-∞, ∞)
What are the vertical asymptotes for the function y = sec(x)?
90°, 270°, 450°, ...
The function y = cot(x) will have a vertical asymptote at 90°.
True
Study Notes
Trigonometric Functions
- Trigonometric functions are used to model real-life situations, such as tides, annual temperatures, and phases of the Moon.
- These functions are periodic, meaning they repeat regularly.
Angles of Any Magnitude
- Angles can be measured in any size, not just acute and obtuse angles.
- Angles can be measured in different quadrants:
- 1st quadrant: 0° to 90°, all ratios are positive
- 2nd quadrant: 90° to 180°, sin is positive, cos and tan are negative
- 3rd quadrant: 180° to 270°, tan is positive, sin and cos are negative
- 4th quadrant: 270° to 360°, cos is positive, sin and tan are negative
- The ASTC rule (All Stations To Central) helps to remember the signs of the ratios in each quadrant.
Trigonometric Ratios
- Trigonometric ratios can be found for angles greater than 360° by turning around the circle more than once.
- Negative angles can be measured in the opposite direction from positive angles.
- The ASTC rule also works for negative angles.
Trigonometric Identities
- Reciprocal trigonometric ratios are the reciprocals of the sine, cosine, and tangent ratios.
- Examples of reciprocal trigonometric ratios:
- sin-1, cos-1, and tan-1
- cosecant, secant, and cotangent (which are the reciprocals of sine, cosine, and tangent respectively)
Terminology
- Amplitude: the height from the center of a periodic function to the maximum or minimum values.
- Centre: the mean value of a periodic function that is equidistant from the maximum and minimum values.
- Period: the length of one cycle of a periodic function.
- Phase: a horizontal shift (translation).
- Identity: an equation that shows the equivalence of two algebraic expressions for all values of the variables.
- Inverse trigonometric functions: the sin-1, cos-1, and tan-1 functions, which are the inverse functions of the sine, cosine, and tangent functions respectively.### Reciprocal Trigonometric Ratios
- The reciprocal trigonometric ratios are: cosecant (cosec), secant (sec), and cotangent (cot)
- The formulas for these ratios are:
- cosec θ = 1 / sin θ
- sec θ = 1 / cos θ
- cot θ = 1 / tan θ
- These ratios have the same signs as their related ratios in the different quadrants
Complementary Angles
- If ∠B = θ, then ∠A = 90° - θ
- Angles ∠B and ∠A are complementary because they add up to 90°
- The trigonometric ratios of complementary angles are equal:
- sin θ = cos (90° - θ)
- cos θ = sin (90° - θ)
- tan θ = cot (90° - θ)
- sec θ = cosec (90° - θ)
- cot θ = tan (90° - θ)
Trigonometric Identities
- The tangent identity: tan θ = sin θ / cos θ
- The Pythagorean identities:
- cos² θ + sin² θ = 1
- 1 + tan² θ = sec² θ
- 1 + cot² θ = cosec² θ
- These identities can be rearranged to give:
- cos² θ = 1 - sin² θ
- sin² θ = 1 - cos² θ
Sums and Differences of Angles
- The formulas for the trigonometric ratios of sums and differences of angles are:
- cos (A - B) = cos A cos B + sin A sin B
- cos (A + B) = cos A cos B - sin A sin B
- sin (A + B) = sin A cos B + cos A sin B
- sin (A - B) = sin A cos B - cos A sin B### Sum and Difference Identities
- The sum and difference identities for trigonometric functions are:
- sin(A + B) = sin A cos B + cos A sin B
- sin(A - B) = sin A cos B - cos A sin B
- cos(A + B) = cos A cos B - sin A sin B
- cos(A - B) = cos A cos B + sin A sin B
- tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
- tan(A - B) = (tan A - tan B) / (1 + tan A tan B)
Double Angle Identities
- The double angle identities for trigonometric functions are:
- sin 2A = 2 sin A cos A
- cos 2A = cos^2 A - sin^2 A
- tan 2A = 2 tan A / (1 - tan^2 A)
Product to Sum Identities
- The product to sum identities for trigonometric functions are:
- cos A cos B = [cos(A + B) + cos(A - B)] / 2
- sin A sin B = [cos(A - B) - cos(A + B)] / 2
- sin A cos B = [sin(A + B) + sin(A - B)] / 2
- cos A sin B = [sin(A + B) - sin(A - B)] / 2
The t-Formulas
- The t-formulas express sin A, cos A, and tan A in terms of t = tan(A/2):
- sin A = 2t / (1 + t^2)
- cos A = (1 - t^2) / (1 + t^2)
- tan A = 2t / (1 - t^2)
Examples and Exercises
- Various examples and exercises are provided to practice and apply the trigonometric identities.
Learn about trigonometric functions, graphs, inverse trigonometric functions, identities, and equations, and how they are used to model real-life periodic situations.
Make Your Own Quizzes and Flashcards
Convert your notes into interactive study material.
Get started for free
