Fill-in Blank Trigonometric Formula Practice Quiz

This is the Pythagorean identity in trigonometry. The formula sin²θ + cos²θ always equals 1 regardless of the angle θ.

This identity directly follows from the Pythagorean identities in trigonometry. The formula clearly shows that 1 plus tan²θ equals sec²θ.

The reciprocal identity for sine states that cscθ is the reciprocal of sinθ. This basic identity reinforces the relationship between a trigonometric function and its reciprocal.

By definition, tangent is the ratio of sine to cosine. This fundamental identity helps in understanding how trigonometric functions relate to each other.

The reciprocal identity for cosine states that secθ equals 1 divided by cosθ. This is an essential identity that is frequently used in trigonometric simplifications.

This is the sine angle sum formula. The missing term is cosα sinβ which, when added to sinα cosβ, gives the full expression for sin(α + β).

The cosine difference formula is given by cos(α - β) = cosα cosβ + sinα sinβ. This structure ensures the correct expansion of the cosine difference.

The double-angle formula for sine states that sin2θ equals 2 sinθ cosθ. This formula is useful in various trigonometric computations and simplifications.

One form of the double-angle formula for cosine is cos2θ = 1 - 2 sin²θ. This identity is valuable for transforming and simplifying trigonometric expressions.

The double-angle formula for tangent is given by tan2θ = (2tanθ)/(1 - tan²θ). This formula expresses the tangent of a double angle in terms of the tangent of a single angle.

The ratio of sine to cosine defines the tangent function. Recognizing this relationship is foundational in trigonometry.

This identity is a rearrangement of the Pythagorean identities. Adding cot²θ to 1 gives csc²θ, reinforcing the reciprocal relationships among trigonometric functions.

Sine is defined as the ratio of the opposite side to the hypotenuse in a right triangle. This is one of the basic definitions in trigonometry.

Using the double-angle identity for sine, sin2θ equals 2 sinθ cosθ, so sinθ cosθ is half of sin2θ. This conversion is useful in simplifying trigonometric expressions.

In the angle sum formula for sine, sinα is multiplied by cosβ. Recognizing this component is key to understanding the complete expansion of sin(α + β).

The identity shows that sinθ + cosθ can be expressed as √2 times sin(θ + 45°). This transformation simplifies the sum of sine and cosine into a single trigonometric function.

Dividing both sides of the equation sinθ = 2cosθ by cosθ gives tanθ = 2. This problem tests the understanding of the relationship between sine, cosine, and tangent.

The half-angle formula for tangent can be expressed as tan(θ/2) = sinθ/(1 + cosθ). Understanding half-angle identities is crucial for solving trigonometric equations.

This identity is derived from the Pythagorean identities and shows that the difference between csc²θ and cot²θ is always 1. It is an important relation in trigonometry.

Substituting tan²θ = 9/16 into the identity yields sec²θ = 1 + 9/16 = 25/16, so secθ equals the square root of 25/16, which is 5/4. This problem requires applying a trigonometric identity to compute the value of secθ.