Trigonometry basics
54 flashcards covering Trigonometry basics for the ACT Math section.
Trigonometry is the study of relationships between the sides and angles of triangles, focusing mainly on right-angled ones. At its core, it uses simple ratios and functions like sine, cosine, and tangent to connect an angle to the lengths of a triangle's sides. For example, sine of an angle is the ratio of the opposite side to the hypotenuse, which helps solve problems involving heights, distances, or shapes in everyday scenarios.
On the ACT Math section, trigonometry questions typically appear in multiple-choice formats, testing your ability to find missing sides or angles in triangles, interpret basic trig functions on graphs, or apply them to word problems like projectile motion. Common traps include confusing sine with cosine, overlooking the Pythagorean theorem, or misapplying functions in non-right triangles. Focus on mastering key identities, practicing with calculators for decimal approximations, and recognizing when to use the unit circle for angle measures.
A concrete tip: Always sketch a triangle to visualize the problem.
Terms (54)
- 01
Sine of an angle
In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
- 02
Cosine of an angle
In a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
- 03
Tangent of an angle
In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
- 04
Cosecant of an angle
The cosecant of an angle is the reciprocal of its sine, equal to the hypotenuse divided by the length of the opposite side in a right triangle.
- 05
Secant of an angle
The secant of an angle is the reciprocal of its cosine, equal to the hypotenuse divided by the length of the adjacent side in a right triangle.
- 06
Cotangent of an angle
The cotangent of an angle is the reciprocal of its tangent, equal to the adjacent side divided by the length of the opposite side in a right triangle.
- 07
Pythagorean theorem
In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.
- 08
SOH-CAH-TOA
A mnemonic for remembering trigonometric ratios: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent.
- 09
Unit circle
A circle with a radius of 1 centered at the origin, used to define trigonometric functions for all angles by their coordinates on the circle.
- 10
Radian measure
A unit for measuring angles based on the radius of a circle, where one radian is the angle subtended by an arc equal to the radius.
- 11
Degree measure
A unit for measuring angles, where a full circle is 360 degrees, commonly used in trigonometry for angles in triangles.
- 12
Trigonometric ratios
The ratios of sides in a right triangle related to an angle, including sine, cosine, tangent, cosecant, secant, and cotangent.
- 13
Exact value of sine 30 degrees
The sine of 30 degrees is exactly 1/2, a standard value derived from a 30-60-90 triangle.
- 14
Exact value of cosine 30 degrees
The cosine of 30 degrees is exactly √3/2, a standard value from a 30-60-90 triangle.
- 15
Exact value of sine 45 degrees
The sine of 45 degrees is exactly √2/2, a standard value from a 45-45-90 triangle.
- 16
Exact value of cosine 45 degrees
The cosine of 45 degrees is exactly √2/2, a standard value from a 45-45-90 triangle.
- 17
Exact value of sine 60 degrees
The sine of 60 degrees is exactly √3/2, a standard value from a 30-60-90 triangle.
- 18
Exact value of cosine 60 degrees
The cosine of 60 degrees is exactly 1/2, a standard value from a 30-60-90 triangle.
- 19
Reference angle
The acute angle formed by the terminal side of an angle and the x-axis, used to find trigonometric values for angles in different quadrants.
- 20
Signs of trigonometric functions in quadrants
Sine is positive in quadrants 1 and 2, cosine is positive in quadrants 1 and 4, and tangent is positive in quadrants 1 and 3.
- 21
Inverse sine function
The inverse of the sine function, which returns the angle whose sine is a given number, with a range from -90 to 90 degrees.
- 22
Inverse cosine function
The inverse of the cosine function, which returns the angle whose cosine is a given number, with a range from 0 to 180 degrees.
- 23
Inverse tangent function
The inverse of the tangent function, which returns the angle whose tangent is a given number, with a range from -90 to 90 degrees.
- 24
Pythagorean identity
The trigonometric identity that states sin²θ + cos²θ = 1 for any angle θ.
- 25
Angle of elevation
The angle formed between the line of sight upward from the horizontal to an object, often used in word problems involving trigonometry.
- 26
Angle of depression
The angle formed between the line of sight downward from the horizontal to an object, commonly appearing in applied trigonometry scenarios.
- 27
Graph of the sine function
The graph of y = sin x is a wave that oscillates between -1 and 1, with a period of 360 degrees or 2π radians, starting at (0,0).
- 28
Graph of the cosine function
The graph of y = cos x is a wave that oscillates between -1 and 1, with a period of 360 degrees or 2π radians, starting at (0,1).
- 29
Amplitude of a trigonometric function
The maximum distance from the midline to the peak or trough of a trigonometric graph, determining the height of the wave.
- 30
Period of a trigonometric function
The length of one complete cycle of a trigonometric graph, such as 360 degrees for sine and cosine without modifications.
- 31
Domain of sine and cosine
The domain of the sine and cosine functions is all real numbers, as they are defined for any angle.
- 32
Range of sine and cosine
The range of the sine and cosine functions is from -1 to 1, inclusive, for all real inputs.
- 33
Even trigonometric functions
Cosine is an even function, meaning cos(-θ) = cos θ, so its graph is symmetric about the y-axis.
- 34
Odd trigonometric functions
Sine and tangent are odd functions, meaning sin(-θ) = -sin θ and tan(-θ) = -tan θ, showing symmetry about the origin.
- 35
Co-function identities
Identities that relate trigonometric functions of complementary angles, such as sin(90° - θ) = cos θ.
- 36
Solving a right triangle
The process of finding all missing sides and angles in a right triangle using trigonometric ratios and the Pythagorean theorem.
- 37
Law of Sines
In any triangle, the ratio of the length of a side to the sine of its opposite angle is constant, useful for non-right triangles.
- 38
Law of Cosines
A formula that relates the lengths of the sides of a triangle to the cosine of one of its angles, generalizing the Pythagorean theorem.
- 39
Trigonometric equation
An equation involving trigonometric functions that requires solving for the angle, such as sin x = 1/2.
- 40
Common mistake with inverse trig functions
Inverse trigonometric functions return angles in a specific range, so solutions may need adjustment for the original equation's context.
- 41
triangle ratios
In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2, with the smallest side opposite the 30-degree angle.
- 42
Vertical asymptotes of tangent
The tangent function has vertical asymptotes at odd multiples of 90 degrees, where it is undefined.
- 43
Phase shift in trig graphs
A horizontal shift of a trigonometric graph, such as in y = sin(x - c), where c is the phase shift amount.
- 44
Vertical shift in trig graphs
A vertical movement of a trigonometric graph, added to the function like y = sin x + k, where k is the shift.
- 45
Sum of angles identity for sine
The formula sin(A + B) = sin A cos B + cos A sin B, used to find sine of the sum of two angles.
- 46
Difference of angles identity for sine
The formula sin(A - B) = sin A cos B - cos A sin B, used for sine of the difference of two angles.
- 47
Double-angle formula for sine
The formula sin(2θ) = 2 sin θ cos θ, which expresses the sine of double an angle in terms of sine and cosine.
- 48
Double-angle formula for cosine
The formula cos(2θ) = cos²θ - sin²θ, or other forms, used to find cosine of double an angle.
- 49
Reciprocal identities
The relationships that csc θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ.
- 50
Quotient identities
The identities that tan θ = sin θ / cos θ and cot θ = cos θ / sin θ.
- 51
Strategy for solving trig word problems
Draw a diagram, identify the right triangle, label known sides and angles, and apply appropriate trigonometric ratios to set up equations.
- 52
Common error with angle measures
Confusing degrees and radians can lead to incorrect calculations, so always check the unit before computing trigonometric values.
- 53
Trigonometric values at 0 degrees
Sine of 0 degrees is 0, cosine is 1, and tangent is 0, representing the starting point on the unit circle.
- 54
Trigonometric values at 90 degrees
Sine of 90 degrees is 1, cosine is 0, and tangent is undefined, as seen on the unit circle.