Trigonometry

Yes because the given dimensions comply with Pythagoras; theorem for a right angle triangle.

csc θ = 1/sin θ

→ sin θ = -1/4

cos² θ + sin² θ = 1

→ cos θ = ± √(1 - sin² θ) = ± √(1 - ¼²) = ± √(1- 1/16) = ± √(15/16) = ± (√15)/4

In Quadrant III both cos and sin are negative

→ cos θ= -(√15)/4

If the angles are the same and the sides are proportional by ratio then they are said to be similar triangles.

1 revolution = 2*pi radianstherefore, k revs per second = 2*k*pi radians per second

or

if you still work in degrees, it is 360*k degrees per second.

Putting more details in the answer means nobody sees the question as unanswered to attempt to answer it.

A sin curve varies evenly either side of the x-axis; the graph varies between -3.5 and +5.5; the mid-point of -3.5 and +5.5 is (5.5 - 3.5) ÷ 2 = 1

→ it has been shifted up the y-axis by 1 unit

→ k = 1

As the curve has been shifted up the y-axis by 1 unit, it varies from (-3.5 -1) = -4.5 to (5.5 - 1) = +4.5

The sin curve varies between -1 and +1

→ it has been stretched by 4.5

→ a = 4.5

Using the points with y-coordinate of 1, the curve is negative between the first two and positive between the second two; thus half the sine curve, (ie between 0 and π radians) is between two of them.

Between the first two of (-5.495, 1) and (0.785, 1) is 0.785 - -5.495 = 6.28 = 2 × 3.14 = 2π

Similarly between the second two of (0.785, 1) and (7.065, 1) is 7.065 - 0.785 = 6.28 = 2 × 3.14 = 2π

→ the x-axis has been stretched by 2π ÷ π = 2

→ b = 2

The sine curve starts increasing from zero, thus (0.785, 1) was originally at the origin (0, 0)

→ the curve has been shifted right by 0.785 = π/4

→ h = π/4

Thus the curve has equation:

y = 4.5 sin (2(x - π/4))

tan θ = sin θ / cos θ

sec θ = 1 / cos θ

sin ² θ + cos² θ = 1 → sin² θ - 1 = - cos² θ

→ tan² θ - sec² θ = (sin θ / cos θ)² - (1 / cos θ)²

= sin² θ / cos² θ - 1 / cos² θ

= (sin² θ - 1) / cos² θ

= - cos² θ / cos² θ

= -1

sin(3π/2) = -1

The given vertices works out as a right angle triangle when plotted on the Cartesian plane and using the distance formula its dimensions with respect to Q are:-

Hypotenuse: 15

Adjacent: 6 times square root of 5

Opposite: 3 times square root of 5

Using any of the 3 trigonometry ratios angle Q works out as in the following:-

sin-1(opp/hyp) = 26.6 degrees to 3 significant figures

tan-1(opp/adj) = 26.6 degrees rounded to 1 decimal place

cos-1(adj/hyp) = 26.6 degrees to the nearest tenth

1/4 of 360 degrees = 90 degrees which is a right angle

The smallest angle of the triangle will be opposite the smallest side and using the cosine rule it works out as 17.9 degrees rounded to one decimal place.

If you mean quadrilateral ABCD then by using Pythagoras' theorem diagonal AC is 5 cm and using the cosine rule angle ADC works out as 41.04 degrees.

Using the sine rule you can find the other two sides:

Final angle = 180° - (37.25° + 48.4°) = 94.35°

→ other two sides are opposite the 37.25° and 48.4° angles as the longest side is opposite the largest angle.

→ the other two sides are:

s1/sin 37.25° = 162mm / sin 94.35° → s1 = 162mm × sin 37.25°/sin 94.35°

and s2 = 162mm × sin 48.4°/sin 94.35°

→ the perimeter = 162mm + 162mm × sin 37.25°/sin 94.35° + 162mm × sin 48.4°/sin 94.35°

= 162mm (1 + (sin 37.25° + sin 48.4°)/sin 94.35°)

≈ 382 mm

The first angle is 166 deg and the second is 14 deg.

It's the statement that in any triangle, the sum of the lengths of any two sides must be greater or equal to the length of the third side.

The question asks about the "indicated operations". In those circumstances would it be too much to expect that you make sure that there is something that is indicated?

csc(x)*{sin(x) + cos(x)} = csc(x)*sin(x) + csc(x)*cos(x)

=1/sin*(x)*sin(x) + 1/sin(x)*cos(x) = 1 + cot(x)

If you mean trigonometry then it is the working properties of triangles.

If those are the only values, no.

You find the smallest positive value y such that

tan(x + y) = tan(x) for all x.

In a right triangle, the sine of one of the angles other than the right angle is the length of the side opposite the angle divided by the length of the hypotenuse (the side opposite the right angle), the cosine is the length of the side adjacent to the angle divided by the length of the hypotenuse, and the tangent is the length of the opposite side divided by the length of the adjacent side.

• tanx = 5
• x = tan-1(5) = arctan5
• x ~ 78.69

There aren't. There are three: Sine, Cosine and Tangent, for any given right-angled triangle.

They are related of course: for any given angle A, sinA/cosA = tanA; sinA + cosA =1.

As you can prove for yourself, the first by a little algebraic manipulation of the basic ratios for a right-angled triangle, the second by looking up the values for any value such that 0 < A < 90.

And those three little division sums are the basis for a huge field of mathematics extending far beyond simple triangles into such fields as harmonic analysis, vectors, electricity & electronics, etc.

Assuming your equation is: e^4x = 2981

Then taking the logs to base e (natural logs) of both sides gives:

ln(e^4x) = ln(2981)

→ 4x = ln(2981)

[Note: The 2981 looks like it has been rounded to a whole number from approx 2980.958 which means the value of x was a whole number (to which ln(2981) rounds).]