Chapter 2: Trigonometric and Inverse Trigonometric Functions (Set-1)
In a right triangle, sin θ is defined as
A Adjacent/Hypotenuse
B Opposite/Adjacent
C Hypotenuse/Opposite
D Opposite/Hypotenuse
For an acute angle θ in a right triangle, sine is the ratio of the side opposite θ to the hypotenuse. This definition links geometry to trigonometric values.
In a right triangle, cos θ equals
A Adjacent/Hypotenuse
B Opposite/Hypotenuse
C Hypotenuse/Adjacent
D Adjacent/Opposite
Cosine of θ is the ratio of the adjacent side to the hypotenuse in a right triangle. It is used widely in projections and basic triangle solving.
In a right triangle, tan θ equals
A Adjacent/Opposite
B Opposite/Hypotenuse
C Opposite/Adjacent
D Adjacent/Hypotenuse
Tangent is defined as opposite divided by adjacent for angle θ in a right triangle. It also equals sin θ / cos θ when cos θ is not zero.
Which identity is always true
A sinx+cosx=1
B sin²x+cos²x=1
C sin²x−cos²x=1
D tanx+cotx=1
The Pythagorean identity comes from the unit circle or right triangle: x²+y²=1. With x=cos x and y=sin x, it gives sin²x + cos²x = 1.
A basic Pythagorean identity is
A 1+tan²x=sec²x
B 1+sin²x=cos²x
C 1+cot²x=cscx
D 1+sec²x=tan²x
Dividing sin²x+cos²x=1 by cos²x gives tan²x+1=sec²x (where cos x ≠ 0). It helps simplify expressions with tan and sec.
Another Pythagorean identity is
A 1+cot²x=sec²x
B 1+sin²x=csc²x
C 1+cot²x=csc²x
D 1+cos²x=sec²x
Dividing sin²x+cos²x=1 by sin²x gives 1+cot²x=csc²x (where sin x ≠ 0). It is useful in proofs and equation solving.
Cofunction identity for acute angles is
A sin(90°−x)=cos x
B sin(90°−x)=sin x
C cos(90°−x)=cos x
D tan(90°−x)=tan x
Complementary angles in a right triangle swap opposite and adjacent sides. That makes sine of the complement equal to cosine of the angle, and similarly for other cofunctions.
Standard unit-circle value of sin 0 is
A 1
B −1
C 0
D Undefined
On the unit circle, angle 0 corresponds to point (1,0). The sine value is the y-coordinate, so sin 0 = 0. This anchors many trig evaluations.
Standard unit-circle value of cos 0 is
A 0
B −1
C Undefined
D 1
At angle 0 on the unit circle, the point is (1,0). Cosine is the x-coordinate, so cos 0 = 1. It is a key starting value for graphs.
Value of tan 45° is
A 1
B 0
C √3
D Undefined
In a 45°–45°–90° triangle, opposite and adjacent legs are equal. So tan 45° = opposite/adjacent = 1. This value is commonly used in simplifications.
Value of sin 30° is
A √3/2
B 1/2
C 1
D 0
In a 30°–60°–90° triangle, the side opposite 30° is half the hypotenuse. Therefore sin 30° = 1/2. It’s a standard exact trig value.
Value of cos 60° is
A 1/2
B √3/2
C 0
D 1
In a 30°–60°–90° triangle, the adjacent side to 60° is half the hypotenuse. Hence cos 60° = 1/2. This supports many identity checks.
Value of sin 90° is
A 0
B −1
C Undefined
D 1
On the unit circle, 90° corresponds to point (0,1). Sine equals the y-coordinate, so sin 90° = 1. It also marks the maximum of sine.
Value of cos 180° is
A 1
B 0
C −1
D Undefined
Angle 180° on the unit circle is point (−1,0). Cosine equals the x-coordinate, so cos 180° = −1. This shows cosine can be negative.
The period of sin x is
A 2π
B π
C π/2
D 4π
The sine function repeats its values after a full rotation around the unit circle. One full rotation is 2π radians, so sin(x+2π)=sin x for all x.
The period of tan x is
A 2π
B π/2
C π
D 4π
Tangent repeats every π radians because tan x = sin x / cos x and both sin and cos change sign after π, keeping the ratio the same. So tan(x+π)=tan x.
Value of csc x is
A 1/sin x
B 1/cos x
C 1/tan x
D sin x/cos x
Cosecant is the reciprocal of sine: csc x = 1/sin x (when sin x ≠ 0). Reciprocal identities help rewrite and simplify trigonometric expressions.
Value of sec x is
A 1/sin x
B 1/cos x
C 1/tan x
D cos x/sin x
Secant is defined as the reciprocal of cosine: sec x = 1/cos x (when cos x ≠ 0). It commonly appears in identities and calculus derivatives.
Value of cot x is
A sin x/cos x
B 1/sin x
C 1/cos x
D cos x/sin x
Cotangent is the reciprocal of tangent, so cot x = cos x/sin x (when sin x ≠ 0). It is useful in trig equations and complementary angle relations.
The identity tan x equals
A cos x/sin x
B 1/sin x
C sin x/cos x
D 1/cos x
Tangent is defined as sin x divided by cos x wherever cos x ≠ 0. This relationship is essential for converting expressions and solving equations in terms of sine and cosine.
sin(−x) equals
A −sin x
B sin x
C −cos x
D cos x
Sine is an odd function. On the unit circle, reflecting angle x to −x flips the y-coordinate sign, so sin(−x)=−sin x. This helps simplify many expressions.
cos(−x) equals
A −cos x
B −sin x
C cos x
D sin x
Cosine is an even function. On the unit circle, changing x to −x keeps the x-coordinate the same, so cos(−x)=cos x. This symmetry is widely used in proofs.
tan(−x) equals
A tan x
B −tan x
C −cot x
D cot x
Tangent is an odd function because tan x = sin x/cos x, and sin is odd while cos is even. Hence tan(−x)=−tan x when defined.
sin(x+2π) equals
A sin x
B cos x
C −sin x
D −cos x
Adding 2π radians means one full revolution on the unit circle. The point returns to the same location, so the sine value repeats: sin(x+2π)=sin x.
cos(x+2π) equals
A sin x
B −cos x
C −sin x
D cos x
Like sine, cosine is periodic with period 2π. One full rotation returns to the same x-coordinate on the unit circle. Therefore cos(x+2π)=cos x for all x.
sin(π−x) equals
A −sin x
B cos x
C sin x
D −cos x
Angles x and (π−x) are supplementary and lie symmetrically in the unit circle with the same y-coordinate. Hence sin(π−x)=sin x, a common transformation identity.
cos(π−x) equals
A −cos x
B cos x
C sin x
D −sin x
For supplementary angles, the x-coordinate changes sign while y-coordinate stays same. Therefore cos(π−x)=−cos x. This helps reduce angles to standard forms.
sin(π+x) equals
A sin x
B −sin x
C −cos x
D cos x
Adding π moves to the opposite point on the unit circle, flipping both coordinates. The y-coordinate changes sign, so sin(π+x)=−sin x. This is used in simplification.
cos(π+x) equals
A cos x
B sin x
C −cos x
D −sin x
At angle (π+x), the unit circle point is opposite to the point at x, so x-coordinate flips sign. Hence cos(π+x)=−cos x, useful in transformation identities.
Double-angle formula for sin 2x is
A 2sin x cos x
B sin²x+cos²x
C 1−2sin²x
D 2cos²x−1
The standard double-angle identity is sin 2x = 2 sin x cos x. It is derived from sin(a+b) by setting a=b=x and is vital for equation solving.
One form of cos 2x is
A 2sin x cos x
B sin²x+cos²x
C 1+tan²x
D cos²x−sin²x
From the addition formula cos(a+b), setting a=b=x gives cos 2x = cos²x − sin²x. It can also be rewritten as 1−2sin²x or 2cos²x−1.
The formula cos 2x equals
A 1−2sin²x
B 1+2sin²x
C 2sin x cos x
D 2tan²x−1
Using cos 2x = cos²x − sin²x and substituting cos²x = 1 − sin²x gives cos 2x = 1 − 2sin²x. This form is useful with sine-only equations.
The formula cos 2x also equals
A 2sin²x−1
B 1+2cos²x
C 2cos²x−1
D 1−2cos²x
Starting with cos 2x = cos²x − sin²x and replacing sin²x with 1−cos²x gives cos 2x = 2cos²x − 1. This form helps in cosine-only equations.
The identity sin(x+y) equals
A sin x sin y + cos x cos y
B cos x cos y − sin x sin y
C cos x sin y − sin x cos y
D sin x cos y + cos x sin y
The sine addition formula is sin(x+y)=sin x cos y + cos x sin y. It is fundamental for deriving double-angle, compound-angle, and product-to-sum identities.
The identity cos(x+y) equals
A cos x cos y + sin x sin y
B cos x cos y − sin x sin y
C sin x cos y + cos x sin y
D sin x sin y − cos x cos y
Cosine addition formula is cos(x+y)=cos x cos y − sin x sin y. It is used for expansions, simplifications, and solving trig equations involving sums of angles.
The identity sin(x−y) equals
A sin x cos y − cos x sin y
B sin x cos y + cos x sin y
C cos x cos y − sin x sin y
D cos x sin y + sin x cos y
Sine difference formula is sin(x−y)=sin x cos y − cos x sin y. It is derived from sin(x+y) by replacing y with −y and using parity.
The identity cos(x−y) equals
A cos x cos y − sin x sin y
B sin x cos y − cos x sin y
C cos x cos y + sin x sin y
D sin x sin y − cos x cos y
Cosine difference formula is cos(x−y)=cos x cos y + sin x sin y. It follows from cos(x+y) by substituting y→−y and using cos(−y)=cos y.
Principal range of sin⁻¹x is
A [−π/2, π/2]
B [0, π]
C (−π, π]
D [0, 2π)
arcsin(x) is defined to give a unique principal value. Its standard principal range is from −π/2 to π/2, where sine is one-to-one, ensuring a proper inverse.
Principal range of cos⁻¹x is
A [−π/2, π/2]
B (−π, π]
C [0, 2π)
D [0, π]
arccos(x) returns the unique principal value in [0, π] because cosine is one-to-one on that interval. This prevents multiple angles giving the same cosine value.
Principal range of tan⁻¹x is
A (−π/2, π/2)
B [0, π]
C [−π/2, π/2]
D (−π, π]
arctan(x) is defined to lie strictly between −π/2 and π/2 where tangent is one-to-one. Endpoints are excluded because tan is undefined at ±π/2.
Domain of sin⁻¹x is
A (−∞, ∞)
B [0, 1]
C [−1, 1]
D (−1, 1)
Since sin θ always lies between −1 and 1, arcsin(x) is defined only for inputs x in [−1,1]. This domain restriction makes inverse sine real-valued.
Domain of cos⁻¹x is
A (−∞, ∞)
B [−1, 1]
C [0, π]
D (−1, 1)
Cosine values also remain within −1 and 1. Therefore arccos(x) is defined for x in [−1,1] only. Outside this interval, real angles cannot produce such cosine values.
Domain of tan⁻¹x is
A (−∞, ∞)
B [−1, 1]
C [0, π]
D (−π/2, π/2)
Tangent can take any real value, so arctan(x) accepts all real numbers as inputs. Its output is restricted to (−π/2, π/2) to make it single-valued.
Simplify sin(sin⁻¹x)
A 1/x
B √(1−x²)
C −x
D x
arcsin(x) returns an angle θ in [−π/2, π/2] with sin θ = x. Applying sin to that angle gives sin(θ)=x, so sin(sin⁻¹x)=x for x in [−1,1].
Simplify cos(cos⁻¹x)
A √(1−x²)
B 1/x
C x
D −x
arccos(x) returns θ in [0,π] such that cos θ = x. Taking cosine again gives cos(θ)=x. Hence cos(cos⁻¹x)=x for x in [−1,1].
Simplify tan(tan⁻¹x)
A x
B 1/x
C √(1+x²)
D −x
arctan(x) produces an angle θ in (−π/2, π/2) with tan θ = x. Applying tan returns tan(θ)=x. So tan(tan⁻¹x)=x for all real x.
For x in [−1,1], cos(sin⁻¹x) equals
A x
B √(1+x²)
C −√(1−x²)
D √(1−x²)
Let θ=sin⁻¹x, so θ is in [−π/2, π/2] and sin θ=x. Then cos θ is nonnegative in this range, and cos θ=√(1−sin²θ)=√(1−x²).
For x in [−1,1], sin(cos⁻¹x) equals
A x
B √(1+x²)
C √(1−x²)
D −√(1−x²)
Let θ=cos⁻¹x with θ in [0,π]. Then cos θ=x and sin θ=√(1−cos²θ)=√(1−x²). Sine is nonnegative on [0,π], so the positive root applies.
Standard relation between sin⁻¹x and cos⁻¹x is
A sin⁻¹x+cos⁻¹x=π/2
B sin⁻¹x−cos⁻¹x=π/2
C sin⁻¹x+cos⁻¹x=π
D sin⁻¹x−cos⁻¹x=0
For x in [−1,1], angles θ=arcsin x and φ=arccos x are complementary because sin θ=x and cos φ=x. Thus θ+φ=π/2, matching principal-value definitions.
If sin x = 0, then x equals
A (2n+1)π/2
B nπ
C 2nπ+π/2
D (2n+1)π
Sine becomes zero at integer multiples of π on the unit circle: 0, π, 2π, etc. Hence the general solution is x = nπ, where n is any integer