The three sides of a triangle are 4,5 and 6.show that cosine of the largest angle is 1/8.hence find the area of the triangle.
To find the cosine of the largest angle, let's use the Law of Cosines, which states that for a triangle with sides a, b, and c and opposite angles A, B, and C respectively, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, let's label the sides as follows:
a = 4
b = 5
c = 6
Let's solve for the cosine of the largest angle, which is opposite side c:
36 = 16 + 25 - 40 * cos(C)
Rearrange the equation:
40 * cos(C) = 41 - 36
40 * cos(C) = 5
Divide both sides by 40:
cos(C) = 1/8
Now, let's find the area of the triangle using the formula:
Area = (1/2) * base * height
We can use Heron's formula to find the area of the triangle, which states that the area can be calculated using the semi-perimeter (s) and the lengths of the sides (a, b, and c) as follows:
Area = sqrt(s * (s - a) * (s - b) * (s - c))
Let's calculate the semi-perimeter:
s = (a + b + c)/2
s = (4 + 5 + 6)/2
s = 15/2 = 7.5
Now, plug in the values into the formula:
Area = sqrt(7.5 * (7.5 - 4) * (7.5 - 5) * (7.5 - 6))
= sqrt(7.5 * 3.5 * 2.5 * 1.5)
= sqrt(82.125)
≈ 9.06
Therefore, the area of the triangle is approximately 9.06 square units.
what a lot of work!
Ok, we found that cosC = 1/8
That means that sinC = √63/8
Now, the area of the triangle is
A = 1/2 ab sinC = 1/2 * 4*5*√63/8 = 5/2 √63
Yes, you are correct. I apologize for the extensive explanation. The area of the triangle can indeed be found using the formula A = 1/2 * ab * sin(C).
By substituting the given values, we have:
A = 1/2 * 4 * 5 * √63/8
= 5/2 * √63/8
= 5/2 * (√63/√64)
= 5/2 * (√63/8)
= 5/2 * (√63/2√2)
= 5/4 * √63
Therefore, the area of the triangle is (5/4) * √63 square units.
Which behavior is both territorial and used for courtship?
sparring or wrestling using antlers or horns
dance rituals with potential partners
defending a nest from predators
baring teeth as a sign of aggression
The behavior that is both territorial and used for courtship is "dance rituals with potential partners."
Birds display a wide variety of courtship behaviors. Which behavior is also a territorial behavior?
singing a particular song to help potential mates hear them
flashing feathers to show their health and grooming abilities
Dancing and calling on a lek (communal ground) with other males, competing for female attention
creating a bower (arch) with interesting objects and defending it from rival males
The behavior that is both a courtship behavior and a territorial behavior is "creating a bower (arch) with interesting objects and defending it from rival males."
Why is climate change a concern for animal reproductive behavior?
Animals might need to spend more time hunting, foraging, and migrating.
Animals might have to change their courtship behavior due to climate change.
Climate change might genetically alter organisms and decrease their reproductive capabilities.
Climate change might reduce or destroy reproductive habitats, eliminating breeding grounds.
All of the listed options are valid reasons why climate change is a concern for animal reproductive behavior.
1. Animals might need to spend more time hunting, foraging, and migrating: Climate change can lead to shifts in temperature, rainfall patterns, and availability of resources, which can affect the availability and distribution of food sources for animals. This can disrupt their reproductive cycles and energy availability for reproduction.
2. Animals might have to change their courtship behavior due to climate change: Changes in climate, such as altered temperature or rainfall patterns, can affect the timing and availability of resources needed for courtship rituals. This can lead to changes in the timing or nature of courtship behavior, affecting successful breeding and reproductive success.
3. Climate change might genetically alter organisms and decrease their reproductive capabilities: Climate change can lead to genetic changes in populations due to natural selection and adaptation to new environmental conditions. These genetic changes can potentially affect reproductive capabilities, fertility, and overall fitness of the species.
4. Climate change might reduce or destroy reproductive habitats, eliminating breeding grounds: Climate change can result in the loss, degradation, or fragmentation of habitats necessary for breeding and reproduction. This can lead to the loss of suitable environments, nesting sites, mating areas, or migration routes, impacting the ability of animals to reproduce successfully.
How does hibernation help animals survive?
Hibernation makes access to food easier.
Hibernation increases the chances of meeting a mate.
Hibernation protects the animal young from predators.
Hibernation reduces the energy required for survival.